Abstract
We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of p-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and \(\mathbb {R}^N\) with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allows us to cover the nonlocal counterparts of a large scope of local diffusion problems like, for example, Stefan problems, Hele–Shaw problems, diffusion in porous media problems and obstacle problems. Nonlinear semigroup theory is the basis for this study.
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1 Introduction and preliminaries
In this article, we study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of p-Laplacian type with nonlinear boundary conditions. The problems are posed in a subset W of a metric random walk space [X, d, m] with a reversible measure \(\nu \) for the random walk m (see Subsect. 1.1 for details). The nonlocal diffusion can hold either in W, in its nonlocal boundary \(\partial _mW\), or in both at the same time. We will assume that \(W\cup \partial _mW\) is m-connected and \(\nu \)-finite. The formulations of the diffusion problems that we study are the following:
and, for nonlinear dynamical boundary conditions,
where \(\gamma \) and \(\beta \) are maximal monotone (multivalued) graphs in \(\mathbb {R}\times \mathbb {R}\), \(\hbox {div}_m\textbf{a}_p\) is a nonlocal Leray–Lions-type operator whose model is the nonlocal p-Laplacian type diffusion operator, and \(\mathcal {N}^{\textbf{a}_p}_\textbf{1}\) is a nonlocal Neumann boundary operator (see Subsect. 2.1 for details). In fact, we solve these problems with greater generality, as we will not only consider them for a set W and its nonlocal boundary \(\partial _m W\), but rather for any two disjoint subsets \(\Omega _1\) and \(\Omega _2\) of X such that their union is m-connected.
These problems can be seen as the nonlocal counterpart of local diffusion problems governed by the p-Laplacian diffusion operator (or a Leray–Lions operator) where two further nonlinearities are induced by \(\gamma \) and \(\beta \) (see, for example, [4, 15] for local problems). In [8], and the references therein, one can find an interpretation of the nonlocal diffusion process involved in this kind of problems. On the nonlinearities (brought about by) \(\gamma \) and \(\beta \), we do not impose any further assumptions aside from the natural one (see Bénilan, Crandall and Sacks [15]):
and (in order for diffusion to take place)
where
Therefore, we work with a rather general class of nonlocal nonlinear diffusion problems with nonlinear boundary conditions. We are able to directly cover: obstacle problems, with unilateral or bilateral obstacles (either in W, in \(\partial _mW\), or in both at the same time); the nonlocal counterpart of Stefan-like problems that involve monotone graphs like the graph inverse of
for \(\lambda >0\); diffusion problems in porous media, where monotone graphs like \(p_s(r)=|r|^{s-1}r\), \(s>0\), are involved; and Hele–Shaw type problems, which involve graphs like
Moreover, if \(\gamma =0\) in problem (1.1), then the dynamics only appear in the nonlocal boundary and we obtain the evolution problem for a nonlocal Dirichlet-to-Neumann operator as a particular case. In addition, the homogeneous Dirichlet boundary condition (\(\beta =\{0\}\times \mathbb {R}\)) and the Neumann boundary condition (\(\beta =\mathbb {R}\times \{0\}\)) are also covered.
Nonlocal diffusion problems of p-Laplacian type involving nonlocal Neumann boundary operators have been recently studied in [43] inspired by the nonlocal Neumann boundary operators for the linear case studied in [29, 35]. Nevertheless, due to the generality of the hypotheses considered in this study, the results that we obtain lead to new existence and uniqueness results, which do not follow from previous works, for a great range of problems. This is true even when the problems are considered on weighted discrete graphs or \(\mathbb {R}^N\) with a random walk induced by a nonsingular kernel, spaces for which only some particular cases of these problems have been studied. Some references are given afterwards. For these ambient spaces and for the precise choice of the nonlocal p-Laplacian operator, Problem (1.1) has the following formulations (see Subsect. 1.1, in particular Examples 1.1 and 1.2, and Definition 1.4, for the necessary definitions and notations):
for weighted discrete graphs, and
for the case of \(\mathbb {R}^N\) with the random walk induced by the nonsingular kernel J. We have detailed these problems with well-known formulations in order to show the extent to which Problems (1.1) and (1.2) cover specific nonlocal problems of great interest.
Nonlinear semigroup theory will be the basis for the study of the existence and uniqueness of solutions of the above problems. This study is developed in Sect. 3, where we prove, as a particular case of Theorem 3.4, the existence of mild solutions of Problem (1.2) for general data in \(L^1\), and of strong solutions assuming extra integrability conditions on the data. Moreover, a contraction and comparison principle is obtained. The same is done for Problem (1.1) in Theorem 3.10. See [9,10,11, 21, 30, 31] and [32], for details on such theory, which is completely covered in the well-known unpublished manuscript Evolution equations governed by accretive operators written by Ph. Bénilan, M. G. Crandall and A. Pazy. A summary of it can be found in [8, Appendix].
To apply the nonlinear semigroup theory, our first aim is to prove the existence and uniqueness of solutions of the problem
for general maximal monotone graphs \(\gamma \) and \(\beta \). This is the nonlocal counterpart of (local) quasilinear elliptic problems with nonlinear boundary conditions (see [5] and [15] for the general study of the local case) and is an interesting problem in itself due to the generality with which we address it. To this aim, we make use of a kind of nonlocal Poincaré-type inequalities (see Appendix A) which help us obtain boundedness arguments. These boundedness arguments together with some monotonicity arguments allow us to prove our results by adapting some of the ideas used in [5] and [15] (see also [7] for a very particular case). The same holds for the diffusion problems. The study of Problem (1.3) is developed in Sect. 2, where we prove, for a more general problem, the existence of solutions (Theorem 2.7) and a contraction and comparison principle (Theorem 2.6). At the end of that section, we deal with another nonlocal Neumann boundary operator.
For linear or quasilinear elliptic problems with boundary conditions, obstacles complicate the existence of solutions. The appearance of this difficulty is better understood when one takes into account the continuity of the solution between the inside of the domain and the boundary via the trace. In fact, for a bounded smooth domain \(\Omega \) in \(\mathbb {R}^N\), \(\gamma \) with bounded domain [0, 1] and \(\beta (r)=0\) for all r, it is not possible to find a weak solution of
for data satisfying \(\varphi \le 0\), \({\widetilde{\varphi }}\le 0\) and \( {\widetilde{\varphi }}\not \equiv 0\) (see [5]). However, in our nonlocal setting this sort of continuity is not present and the study of these nonlocal diffusion problems with obstacles hence differs from the study of the local ones (see [6] for a detailed study of these local problems). In particular, we do not need to impose any assumptions on the nonlinearities \(\gamma \) and \(\beta \) aside from the natural ones.
There is a very long list of references for the local elliptic and parabolic counterparts of the problems that we study; see, for example, [4, 5, 11,12,13, 15, 24, 44, 45], and the references therein. See also [38] for a Hele–Shaw problem with dynamical boundary conditions and the references therein. For some particular nonlocal problems we refer to [7, 8, 16, 19, 23, 36, 39, 43]. For fractional diffusion problems, we refer, for example, to [40], where Dirichlet and Neumann boundary conditions are considered; to [17, 18, 25, 26, 34], where fractional porous medium equations are studied, see also J. L. Vázquez’s survey [46] and the references therein; and to [27, 28] for fractional diffusion problems for the Stefan problem.
We now introduce the framework space considered and some other concepts that will be used later on.
1.1 Metric random walk spaces
Let (X, d) be a Polish metric space equipped with its Borel \(\sigma \)-algebra. In the following, whenever we consider a measure on X we assume that it is defined on this \(\sigma \)-algebra.
As introduced in [47], a random walk m on X is a family of Borel probability measures \(m_x\) on X, \(x \in X\), satisfying the two technical conditions: (1) the measures \(m_x\) depend measurably on the point \(x \in X\), i.e., for any Borel set A of X and any Borel set B of \({\mathbb {R}}\), the set \(\{ x \in X:m_x(A) \in B \}\) is Borel; (2) each measure \(m_x\) has finite first moment, i.e., for some (hence any) \(z \in X\), and for any \(x \in X\) one has \(\int _X d(z,y) dm_x(y) < +\infty \).
A metric random walk space [X, d, m] is a Polish metric space (X, d) together with a random walk m.
A \(\sigma \)-finite measure \(\nu \) on X is invariant with respect to the random walk \(m=(m_x)\) if
Moreover, the measure \(\nu \) is said to be reversible with respect to m if the following balance condition holds:
that is, for any Borel set \(C \subset X \times X\),
Under suitable assumptions on the metric random walk space [X, d, m], such a reversible measure \(\nu \) exists and is unique. Note that the reversibility condition implies the invariance condition.
Assumption 1
From this point onwards, [X, d, m] is a metric random walk space equipped with a \(\sigma \)-finite measure \(\nu \) which is reversible (thus invariant) with respect to m.
Let \(\mathcal {B}\) be the Borel \(\sigma \)-algebra of (X, d). Since \(\nu \) is a \(\sigma \)-finite measure on \((X,\mathcal {B})\) and m is a stochastic kernel on \((X,\mathcal {B})\), we may define the tensor product \(\nu \otimes m_x\) of \(\nu \) and m (see, for example, [33, Section 1.2.2], see also [1, Section 2.5]), which is a measure on \((X\times X, \mathcal {B}\otimes \mathcal {B})\), by
Then, a \(\sigma \)-finite measure \(\nu \) invariant with respect to m is reversible if, and only if, the measure \(\nu \otimes m_x\) is symmetric. Note that, for every \(g\in L^1(X\times X,\nu \otimes m_x)\),
Example 1.1
An important class of examples of metric random walk spaces is composed by those which are obtained from weighted discrete graphs. Let \(G = (V(G), E(G),(w_{xy})_{x,y\in V(G)})\) be a weighted discrete graph, where V(G) is the set of vertices, E(G) is the set of and \(w_{xy} = w_{yx}\) is the nonnegative weight assigned to the edge \((x,y) \in E(G)\). We suppose that \(w_{xy} = 0\) if \((x,y) \not \in E(G)\) for \(x,y\in V(G)\). In this case, the following probability measures define a random walk on \((V(G),d_G)\) (here, \(d_G\) is the standard graph distance):
where \(d_x:= \sum _{y\sim x} w_{xy} = \sum _{y\in V(G)} w_{xy}\). Note that, if \(w_{x,y}=1\) for every \((x,y)\in E(G)\), then \(d_x\) coincides with the degree of the vertex x in the graph, that is, the number of edges containing the vertex x. Moreover, the measure \(\nu _G\) defined by
is a reversible measure with respect to this random walk.
Example 1.2
Another important class of examples is given by those of the form \([{\mathbb {R}}^N, d,m^J]\) where d is the Euclidean distance and \(m^J\) is defined as follows: let \(J:{\mathbb {R}}^N\rightarrow [0,+\infty [\) be a measurable, nonnegative and radially symmetric function satisfying \(\int _{{\mathbb {R}}^N}J(z){\text {d}}\mathcal {L}^N(z)=1\) (\(\mathcal {L}^N\) is the Lebesgue measure) and set
In this case, \(\mathcal {L}^N\) is a reversible measure with respect to this random walk.
See [41] (in particular [41, Example 1.2]) for a more detailed exposition of these and other examples.
Definition 1.3
Given two measurable subsets A, \(B \subset X\), we define the m-interaction between A and B as:
Note that, whenever \(L_m(A,B) < +\infty \), if \(\nu \) is reversible with respect to m,
Definition 1.4
Given a measurable set \(\Omega \subset X\), we define its m-boundary as:
and its m-closure as:
Moreover, we define the following ergodicity property.
Definition 1.5
Let [X, d, m] be a metric random walk space with a reversible measure \(\nu \) with respect to m, and let \(\Omega \subset X\) be a measurable and non-\(\nu \)-null subset. We say that \(\Omega \) is m-connected if \(L_m(A,B)>0\) for every pair of measurable non-\(\nu \)-null sets A, \(B\subset \Omega \) such that \(A\cup B=\Omega \) (see [41]).
We recall the following nonlocal notions of gradient and divergence.
Definition 1.6
Given a function \(u: X \rightarrow {\mathbb {R}}\) we define its nonlocal gradient \(\nabla u: X \times X \rightarrow {\mathbb {R}}\) as:
For a function \(\textbf{z}: X \times X \rightarrow {\mathbb {R}}\), its m-divergence \(\textrm{div}_m \textbf{z}: X \rightarrow {\mathbb {R}}\) is defined as:
1.2 Yosida approximation and a Bénilan–Crandall relation
Given a maximal monotone graph \(\vartheta \) in \({\mathbb {R}}\times {\mathbb {R}}\) (see [21]) and \(\lambda >0\), let us denote by
the Yosida approximation of \(\vartheta \) of parameter \(1/\lambda \).
The function \(\vartheta _\lambda \) is maximal monotone and Lipschitz continuous with Lipschitz constant \(\lambda \) (see [21, Proposition 2.6]. Moreover, \(\lim _{\lambda \rightarrow +\infty } \vartheta _\lambda (s) = \vartheta ^0 (s)\) where
is an extension to \({\mathbb {R}}\) of the minimal section of \(\vartheta \). Furthermore, if \(s\in D(\vartheta )\), \(|\vartheta _\lambda (s)|\le |\vartheta ^0(s)|\) for every \(\lambda >0\), and \(|\vartheta _\lambda (s)|\) is nondecreasing in \(\lambda \).
Given a maximal monotone graph \(\vartheta \) in \({\mathbb {R}}\times {\mathbb {R}}\) with \(0\in \vartheta (0)\), we define, for \(s\in D(\vartheta )\),
and
Note that the Yosida approximation \((\vartheta _+)_\lambda \) of \(\vartheta _+\) is nondecreasing in \(\lambda >0\) and \((\vartheta _-)_\lambda \) is nonincreasing in \(\lambda >0\). Observe also that \((\vartheta _+)_\lambda (s)=0\) for \(s\le 0\) and \((\vartheta _-)_\lambda (s)=0\) for \(s\ge 0\), for every \(\lambda >0\), and \(\vartheta _++\vartheta _-=\vartheta \).
Given a maximal monotone graph \(\vartheta \) with \(0\in D(\vartheta )\), \(j_\vartheta (r):=\int _0^r\vartheta ^0(s){\text {d}}s\), \(r\in {\mathbb {R}}\), defines a convex and lower semicontinuous function such that \(\vartheta \) is equal to the subdifferential of \(j_\vartheta \):
Moreover, if \(j_\vartheta {}^*\) is the Legendre transform of \(j_\vartheta \), then
We now recall a Bénilan–Crandall relation between functions \(u, v\in L^1(\Omega ,\nu )\). Denote by \(J_0\) and \(P_0\) the following sets of functions:
Assume that \(\nu (\Omega ) < +\infty \) and let \(u,v\in L^1(\Omega ,\nu )\). The following relation between u and v is defined in [14]:
Moreover, the following equivalences are proved in [14, Proposition 2.2] (we only give the particular cases that we use):
2 Nonlocal stationary problems
In this section, we give our main results concerning the existence and uniqueness of solutions of the nonlocal stationary Problem (1.3). We start by recalling the class of nonlocal Leray–Lions-type operators and the Neumann boundary operators that we will be working with, which were introduced in [43].
2.1 Nonlocal diffusion operators of Leray–Lions-type and nonlocal Neumann boundary operators
For \(1<p<+\infty \), let us consider a function \(\textbf{a}_p:X\times X\times \mathbb {R}\rightarrow \mathbb {R}\) such that
there exist constants \(c_p,C_p>0\) such that
and
Condition (2.2) and the last condition imply that
for \(\nu \otimes m_x\)-a.e. \((x,y)\in X \times X\hbox { and for every } r\in {\mathbb {R}}\).
For \(u:X\rightarrow \mathbb {R}\), let us define \(\textbf{z}_{\textbf{a}_p,u}:X\times X\rightarrow {\mathbb {R}}\) by
Then, by Definition 1.6 and on account of (2.2),
For simplicity, we write
An example of a function \(\textbf{a}_p\) satisfying the above assumptions is
where \(\varphi :X\rightarrow {\mathbb {R}}\) is a measurable function satisfying \(0<{c}\le \varphi \le {C}\), where c and C are constants. In particular, if \(\varphi (x)=2\) for every \( x\in X\),
is the (nonlocal) p-Laplacian operator on the metric random walk space [X, d, m].
Observe that \(\hbox {div}_m \textbf{a}_p u(x)\) defines a kind of Leray–Lions operator for the random walk m.
We now recall the nonlocal Neumann boundary operators introduced in [43]. Let us consider a measurable set \(W\subset X\) with \(\nu (W)>0\). The Gunzburger–Lehoucq-type Neumann boundary operator on \(\partial _mW\) is given by
where, taking into account the supports of the \(m_x\), we have that in fact, the integral is being calculated over the nonlocal tubular boundary \(\partial _mW\cup \partial _m(X\setminus W)\) of W. On the other hand, the Dipierro–Ros-Oton–Valdinoci-type Neumann boundary operator on \(\partial _mW\) is given by
for which, in this case, the integral is being calculated over the nonlocal boundary \(\partial _m(X\setminus W)\) of \(X\setminus W\).
For each of these Neumann boundary operators and for \(\varphi \) defined on \(W_m=W \cup \partial _m W\), we can look for solutions of the following problem:
\(\textbf{j}\in \{1,2\}\). Observe that, by the reversibility of \(\nu \) with respect to m and recalling the definitions of \(\partial _m W\) and \(W_m\) (Definition 1.4), \(m_x(X{\setminus } W_m)=0\) for \(\nu \)-a.e. \(x\in W\). Indeed,
Consequently,
Lemma 2.1
Let \(\Omega \subset X\) be a \(\nu \)-finite set and let \(\{u_k\}_{k\in {\mathbb {N}}}\subset L^p(\Omega ,\nu )\) such that \(u_k{\mathop {\longrightarrow }\limits ^{k}} u\in L^p(\Omega ,\nu )\) in \(L^p(\Omega ,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega \). Suppose also that there exists \(h\in L^p(\Omega ,\nu )\) such that \(|u_k|\le h\) \(\nu \)-a.e. in \(\Omega \). Then,
and, in particular,
Taking a subsequence if necessary, the \(\nu \)-a.e. pointwise convergence and the domination by the function h in the hypotheses are a consequence of the convergence in \(L^p(\Omega ,\nu )\).
Proof
Let \(A\subset \Omega \) be a \(\nu \)-null set such that \(|u_k(x)|\le h(x)<+\infty \) for every \(x\in \Omega {\setminus } A\) and every \(k\in {\mathbb {N}}\), and such that \(u_k(x){\mathop {\longrightarrow }\limits ^{k}} u(x)\) for every \(x\in \Omega \setminus A\). By (2.1), there exists a \(\nu \otimes m_x\)-null set \(N_1 \subset \Omega \times \Omega \) such that \(\textbf{a}_p(x,y,\cdot )\) is continuous for every \((x,y)\in (\Omega \times \Omega ){\setminus } N_1\). Therefore,
for every \((x,y)\in (\Omega \times \Omega ){\setminus } (N_1\cup (A\times \Omega )\cup (\Omega \times A))\), where, by the reversibility of \(\nu \) with respect to m, \(N_1\cup (A\times \Omega )\cup (\Omega \times A)\) is also \(\nu \otimes m_x\)-null. Moreover, by (2.4), there exists a \(\nu \otimes m_x\)-null set \(N_2 \subset \Omega \times \Omega \) such that
for every \((x,y)\in (\Omega \times \Omega ){\setminus } (N_2\cup (A\times \Omega )\cup (\Omega \times A))\) and some constant \(\widetilde{C}\), where, again, \(N_2\cup (A\times \Omega )\cup (\Omega \times A)\) is \(\nu \otimes m_x\)-null. Then, taking \((x,y)\in (\Omega \times \Omega ){\setminus }(N_1\cup N_2\cup (A\times \Omega )\cup (\Omega \times A))\),
and
Now, by the invariance of \(\nu \) with respect to m, since \(h\in L^{p}(\Omega ,m_x)\) and \(\nu (\Omega )<+\infty \), we have that for \({{\tilde{h}}}(x,y):= 1+|h(x)|^{p-1}+|h(y)|^{p-1}\), \({\tilde{h}}\in L^{p'}(\Omega \times \Omega ,\nu \otimes m_x)\), so we may apply the dominated convergence theorem to conclude. \(\square \)
2.2 Existence and uniqueness of solutions of doubly nonlinear stationary problems under nonlinear boundary conditions
As mentioned in the introduction, the aim here is to study the existence and uniqueness of solutions of the problem
where \(W\subset X\) is m-connected and \(\nu (W_m)<+\infty \). See [5, 15] for the reference local models. In Subsect. 2.3, we address this problem but with the nonlocal Neumann boundary operator \(\mathcal {N}^{\textbf{a}_p}_2\) instead.
Problem (2.7) is a particular case (recall (2.6)) of the following general, and interesting by itself, problem. Let \(\Omega _1,\Omega _2\subset X\) be disjoint measurable non-\(\nu \)-null sets and let
Given \(\varphi \in L^{1}(\Omega ,\nu )\), we consider the problem
For simplicity, we generally use the notation \((GP_\varphi )\) in place of \((GP_\varphi ^{ \textbf{a}_p,\gamma ,\beta })\). However, we use the more detailed notation further on. Moreover, we make the following assumptions.
Assumption 2
We assume that \(\Omega =\Omega _1\cup \Omega _2\) is m-connected and \(\nu (\Omega )<+\infty \).
Remark 2.2
Observe that, given an m-connected set \(\Omega \subset X\) (recall Definition 1.5), \(m_x(\Omega )>0\) for \(\nu \)-a.e. \(x\in \Omega \). Indeed, if
then
thus \(\nu (N)=0\).
Assumption 3
Let
where the notation 
means that 
and 
are mutually singular. We assume that
Remark 2.3
Note that, for \(x\in \Omega \) such that \(m_x(\Omega )>0\), if \(m_x\ll \nu \) (i.e., \(m_x\) is absolutely continuous with respect to \(\nu \), do not confuse the use of \(\ll \) in this context with its use in the notation in Subsect. 1.2) then 
. Therefore, by Remark 2.2, if \(m_x\ll \nu \) for \(\nu \)-a.e. \(x\in \Omega \) then \(\nu \left( \mathcal {N}_\perp ^\Omega \right) =0\). Hence, the above condition is weaker than assuming that \(m_x\ll \nu \) for \(\nu \)-a.e. \(x\in \Omega \).
Assumption 4
We assume, together with \(0\in \gamma (0)\cap \beta (0)\), that
where
Assumption 5
We assume that the following generalised Poincaré type inequality holds: For every \(0<l\le \nu (\Omega )\), there exists a constant \(\Lambda >0\) such that, for every \(u \in L^p(\Omega ,\nu )\) and any measurable set \(Z\subset \Omega \) with \(\nu (Z)\ge l\),
This assumption holds true in many important examples (see Appendix A).
From now on in this subsection, we work under Assumptions 1 to 5.
Definition 2.4
A solution of \((GP_\varphi )\) is a pair [u, v] with \(u\in L^p(\Omega ,\nu )\) and \(v\in L^{p'}(\Omega ,\nu )\) such that
-
1.
\(v(x)\in \gamma (u(x))\ \hbox { for}~\nu \hbox {-a.e. } x\in \Omega _1,\)
-
2.
\(v(x)\in \beta (u(x))\ \hbox { for}~\nu \hbox {-a.e. } x\in \Omega _2,\)
-
3.
\([(x,y)\mapsto a_p(x,y,u(y)-u(x))]\in L^{p'}(\Omega \times \Omega ,\nu \otimes m_x)\),
-
4.
and
$$\begin{aligned} v(x) - \int _{\Omega } \textbf{a}_p(x,y,u(y)-u(x)) dm_x(y)=\varphi (x), \quad x \in \Omega . \end{aligned}$$
A subsolution (supersolution) of \((GP_\varphi )\) is a pair [u, v] with \(u\in L^p(\Omega ,\nu )\) and \(v\in L^1(\Omega ,\nu )\) satisfying 1., 2., 3. and
Remark 2.5
(Integration by parts formula) The following integration by parts formula which results from the reversibility of \(\nu \) with respect to m, can be easily proved. Let u be a measurable function such that
and let \(w \in L^{q'}(\Omega ,\nu )\). Then,
Let us see, formally, the way in which we use the above integration by parts formula in what follows. Suppose that we are in the following situation:
Then, multiplying both equations by a test function w, integrating them with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, adding them and using the integration by parts formula we get
Moreover, as a consequence of (2.3), taking \(u=u_i\), \(f=f_i\) and \(g=g_i\), \(i=1,2\), in the above system and for every nondecreasing function \(T:\mathbb {R}\rightarrow \mathbb {R}\), we obtain
The next result gives a maximum principle for solutions of Problem \((GP_\varphi )\) given in (2.8) and, consequently, also for solutions of Problem (2.7).
Theorem 2.6
(Contraction and comparison principle) Let \(\varphi _1\), \(\varphi _2\in L^{1}(\Omega ,\nu )\). Let \([u_{1},v_1]\) be a subsolution of \((GP_{\varphi _1})\) and \([u_{2},v_2]\) be a supersolution of \((GP_{\varphi _2})\). Then,
Moreover, if \(\varphi _1\le \varphi _2\) with \(\varphi _1\ne \varphi _2\), then \(v_1\le v_2\), \(v_1\ne v_2\), and \(u_1\le u_2\) \(\nu \)-a.e. in \(\Omega \).
Furthermore, if \(\varphi _1= \varphi _2\) and \([u_i,v_i]\) is a solution of \((GP_{\varphi _i})\), \(i=1,2\), then \(v_1=v_2\) \(\nu \)-a.e. in \(\Omega \) and \(u_1-u_2\) is \(\nu \)-a.e. equal to a constant.
Proof
By hypothesis,
for \(x \in \Omega \). Let \(k>0\) and \(T_k:{\mathbb {R}}\rightarrow [-k,k]\) be the truncation operator defined as:
and denote \(T_k^+(s):=(T_k(s))^+\). Multiplying the above inequality by \(\frac{1}{k} T_k^+(u_{1}-u_{2}+ k \, \hbox {sign}_0^+(v_1-v_2))\) and integrating over \(\Omega \), we get
Moreover, by the integration by parts formula (Remark 2.5),
Now, since the integrand on the right-hand side is bounded from below by an integrable function, we can apply Fatou’s lemma to get (recall the last observation in Remark 2.5)
Hence, taking limits in (2.11), we get
and (2.9) is proved.
Take now \(\varphi _1\le \varphi _2\) with \(\varphi _1\ne \varphi _2\), then, by (2.9), \(v_1\le v_2\) \(\nu \)-a.e. in \(\Omega \). Now, since \([u_{1},v_1]\) is a subsolution of \((GP_{\varphi _1})\)
thus
Therefore, with the same calculation for \([u_{2},v_2]\),
thus \(v_1\ne v_2\). Now, since \((\varphi _1-\varphi _2)^+=0\) and \((v_1-v_2)^+=0\), from (2.11) we get that
However, since \(v_i(x)\in \gamma (u_i(x))\) for \(\nu \)-a.e. \(x\in \Omega _1\) and \(v_i(x)\in \beta (u_i(x))\) for \(\nu \)-a.e. \(x\in \Omega _2\), \(i=1,2\), we get, by the monotonicity of the graphs, \(u_1(x)\le u_2(x)\) for \(\nu \)-a.e. \(x\in \Omega \) such that \(v_1(x)<v_2(x)\). Therefore, \(\left( v_1(x)-v_2(x)\right) \frac{1}{k} T_k^+(u_{1}(x)-u_{2}(x)))=0\) for \(\nu \)-a.e. \(x\in \Omega \) and thus
Now, recalling Remark 2.5 (that is, integration by parts), we obtain
and thus
for \((x,y)\in (\Omega \times \Omega )\setminus N\) where \(N\subset \Omega \times \Omega \) is a \(\nu \otimes m_x\)-null set. Let \(C\subset \Omega \) be a \(\nu \)-null set such that the section \(N_x:=\{y\in \Omega \,:\, (x,y)\in N\}\) of N is \(m_x\)-null for every \(x\in \Omega {\setminus } C\) and let us see that \(u_1(x)\le u_2(x)\) for every \(x\in \Omega \setminus (C\cup \mathcal {N}_\perp ^\Omega )\) (recall Assumption 3 for the definition of the \(\nu \)-null set \(\mathcal {N}_\perp ^\Omega \)). Suppose that there exists \(x_0\in \Omega \setminus (C\cup \mathcal {N}_\perp ^\Omega )\) such that \(u_1(x_0)-u_2(x_0)>0\). Then, from (2.12) (and (2.3)), we get that \(u_1(y)-u_2(y)=u_1(x_0)-u_2(x_0)>0\) for every \(y\in \Omega \setminus N_{x_0}\). Let
Since \(x_0\not \in \mathcal {N}_\perp ^\Omega \) and \(m_{x_0}(N_{x_0})=0\), we must have \(\nu (S)\ge \nu (\Omega {\setminus } N_{x_0})>0\). Now, following the same argument as before, if \(x\in S\), then \(\Omega \setminus N_x\subset S\) thus \(m_x(\Omega {\setminus } S)\le m_x(N_x)=0\) and, therefore,
However, since \(\Omega \) is m-connected and \(\nu (S)>0\), we must have \(\nu (\Omega \setminus S)=0\); thus, \(u_1(y)-u_2(y)=u_1(x_0)-u_2(x_0)>0\) for \(\nu \)-a.e. \(y\in \Omega \). This contradicts that \(v_1\le v_2\), \(v_1\ne v_2\), \(\nu \)-a.e. in \(\Omega \).
Finally, suppose that \([u_{1},v_1]\) and \([u_{2},v_2]\) are solutions of \((GP_{\varphi })\) for some \(\varphi \in L^1(\Omega ,\nu )\). Then,
thus, since \(v_1=v_2\) \(\nu \)-a.e. in \(\Omega \),
Multiplying this equation by \(u_1-u_2\), integrating over \(\Omega \) and using the integration by parts formula as in Remark 2.5 we get
thus, by (2.3) and positivity,
for \((x,y)\in (\Omega \times \Omega )\setminus N'\) where \(N'\subset \Omega \times \Omega \) is a \(\nu \otimes m_x\)-null set. Let \(C'\subset \Omega \) be a \(\nu \)-null set such that the section \(N'_x:=\{y\in \Omega \,:\, (x,y)\in N'\}\) of \(N'\) is \(\nu \)-null for every \(x\in \Omega \setminus C'\), and let us see that there exists \(L\in {\mathbb {R}}\) such that \(u_1(x)- u_2(x)=L\) for \(\nu \)-a.e. \(x\in \Omega \). Let \(x_0\in \Omega \setminus C'\), \(L:=u_1(x_0)- u_2(x_0)\) and
By (2.13), \(\Omega \setminus C'_{x_0}\subset S'\). Proceeding as we did before to prove that \(\nu (\Omega \setminus S)=0\), we obtain that \(\nu (\Omega \setminus S')=0\). \(\square \)
In order to prove the existence of solutions of Problem (2.8) (Theorem 2.7), we first prove the existence of solutions of an approximate problem. Then, we obtain some monotonicity and boundedness properties of the solutions of these approximate problems that allow us to pass to the limit. This method lets us get around the loss of compactness results in our setting with respect to the local setting. Indeed, we follow ideas used in [5], but, as we have said, making the most of the monotonicity arguments since the Poincaré-type inequalities here only produce boundedness in \(L^{p}\) spaces (versus the boundedness in \(W^{1,p}\) spaces obtained in their local setting). This will be done in the following subsections.
2.2.1 Existence of solutions of an approximate problem
Take \(\varphi \in L^{\infty }(\Omega ,\nu )\). Let \(n, k\in \mathbb {N}\), \(K>0\) and
be defined by
where
for \(x\in \Omega _1\), and
for \(x\in \Omega _2\). Here, \(T_K\) is the truncation operator defined in (2.10) and \((\gamma _+)_k\), \((\gamma _-)_n\), \((\beta _+)_k\) and \((\beta _-)_n\) are Yosida approximations as defined in Subsect. 1.2.
It is easy to see that A is continuous and, moreover, it is monotone and coercive in \(L^p(\Omega , \nu )\). Indeed, the monotonicity results from the integration by parts formula (Remark 2.5) and the coercivity results from the following computation (where the term involving \(\textbf{a}_p\) has been neglected because it is nonnegative, as shown in Remark 2.5):
Therefore, since \(\varphi \in L^{\infty }(\Omega ,\nu )\subset L^{p'}(\Omega ,\nu )\), by [20, Corollary 30], there exist \(u_{n,k}\in L^{p}(\Omega , \nu )\), n, \(k\in {\mathbb {N}}\), such that
That is,
and
Let n, \(k\in {\mathbb {N}}\). We start by proving that \(u_{n,k}\in L^\infty (\Omega ,\nu )\). Set
Then, multiplying (2.14) and (2.15) by \((u_{n,k}-M)^+\), integrating over \(\Omega _1\) and \(\Omega _2\), respectively, adding both equations and neglecting the terms which are zero, we get
Now, by the integration by parts formula (recall Remark 2.5),
Hence, neglecting nonnegative terms in (2.16), we get
thus
Now, subtracting \(\displaystyle \int _\Omega M^{p-1}(u_{n,k}(x)-M)^+{\text {d}}\nu (x)\) from both sides of the above inequality yields
and, consequently, taking \(K>M\), we get
Similarly, taking \(w=(u_{n,k}+M)^-\), we get
which yields, taking also \(K>M\),
Therefore,
as desired.
Now, taking
equations (2.14) and (2.15) yield
and
Take now \(\varphi \in L^{p'}(\Omega ,\nu )\) and, for \(n,k\in \mathbb {N}\), set
Then, since \(\varphi _{n,k}\in L^\infty (\Omega ,\nu )\), by the previous computations leading to (2.17) and (2.18), there exists a solution \(u_{n,k}\in L^\infty (\Omega ,\nu )\) of the following approximate problem (2.20)–(2.21):
Moreover, we obtain the following estimates which will be used later on. Multiplying (2.20) and (2.21) by \(\frac{1}{s} T_s(u_{n,k}^+)\), integrating with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, adding both equations, applying the integration by parts formula (Remark 2.5), and letting \(s\downarrow 0\), we get, after neglecting some nonnegative terms, that
Similarly, multiplying by \(\frac{1}{s} T_s(u_{n,k}^-)\), we get
2.2.2 Monotonicity of the solutions of the approximate problems
Using that \(\varphi _{n,k}\) is nondecreasing in n and nonincreasing in k, and thanks to the way in which we have approximated the maximal monotone graphs \(\gamma \) and \(\beta \), we obtain monotonicity properties for the solutions of the approximate problems.
Fix \(k\in {\mathbb {N}}\). Let \(n_1<n_2\). Multiply equations (2.20) and (2.21) with \(n=n_1\) by \((u_{n_1,k}-u_{n_2,k})^+\), integrate with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, and add both equations. Then, doing the same with \(n=n_2\) and subtracting the resulting equation from the one that we have obtained for \(n=n_1\), we get
Since \((\gamma _+)_k\) and \((\beta _+)_k\) are maximal monotone, the first and third summands on the left-hand side are nonnegative, and the same is true for the second and fourth summands since \((\gamma _-)_{n_1}\ge (\gamma _-)_{n_2}\), \((\beta _-)_{n_1}\ge (\beta _-)_{n_2}\) and these are all maximal monotone. The fifth summand is also nonnegative as illustrated in Remark 2.5. Then, since the last two summands are obviously nonnegative, we get that, in fact,
and
which together imply that
Similarly, we obtain that, for a fixed n, \(u_{n,k}\) is \(\nu \)-a.e. in \(\Omega \) nonincreasing in k.
2.2.3 An \(L^{p}\)-estimate for the solutions of the approximate problems
Multiplying (2.20) and (2.21) by
integrating with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, adding both equations and using the integration by parts formula (Remark 2.5) we get
For the first summand on the left-hand side of (2.24), we have
and for the second
Since \( F_{n,k} (s):=\frac{1}{n} |s|^{p-2}s^+-\frac{1}{k} |s|^{p-2}s^-\) is nondecreasing, for the fourth summand on the left-hand side of (2.24) we have that
Finally, recalling (2.5) for the third summand in (2.24), we get
Now, by Hölder’s inequality and the generalised Poincaré-type inequality with \(l=\nu (\Omega _1)\) (let \(\Lambda _1\) denote the constant appearing in the generalised Poincaré-type inequality in Assumption 5),
and, by (2.22), (2.23) and the generalised Poincaré-type inequality with \(l=\nu (\Omega _1)\) and with \(l=\nu (\Omega _2)\) (let \(\Lambda _2\) denote the constant appearing in the Poincaré-type inequality for the latter case), we obtain
Therefore, by (2.24) and the subsequent equations,
2.2.4 Existence of solutions of \((GP_\varphi )\)
Observe that a solution (u, v) of \((GP_\varphi )\) satisfies
therefore, since \(v\in \gamma (u)\) in \(\Omega _1 \) and \(v\in \beta (u)\) in \(\Omega _2\), we need \(\varphi \) to satisfy
We prove the existence of solutions when the inequalities in the previous equation are strict. This suffices for what we need in the next section. Recall that we are working under the Assumptions 1 to 5.
Theorem 2.7
Given \(\varphi \in L^{p'}(\Omega ,\nu )\) such that
Problem \((GP_\varphi )\) stated in (2.8) has a solution.
Observe then that any solution (u, v) of \((GP_\varphi )\) under such assumptions also satisfies
This will be used later on.
We divide the proof into three cases.
Proof
(Proof of Theorem 2.7 when \(\mathcal {R}_{\gamma ,\beta }^\pm =\pm \infty \)) Suppose that
Let \(\varphi \in L^{p'}(\Omega ,\nu )\), \(\varphi _{n,k}\) defined as in (2.19) and let \(u_{n,k}\in L^\infty (\Omega ,\nu )\), \(n, k\in {\mathbb {N}}\), be solutions of the Approximate Problem (2.20)–(2.21).
Step A (Boundedness). Let us first see that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) is bounded.
Step 1. We start by proving that \(\{\Vert u_{n,k}^+\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) is bounded. We see this case by case. Since \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \), then \(\sup \text{ Ran }(\gamma )=+\infty \) or \(\sup \text{ Ran }(\beta )=+\infty \).
Case 1.1. Suppose that \(\sup \text{ Ran }(\gamma )=+\infty \). Then, by (2.22),
Let \(z^+_{n,k}:= (\gamma _+)_k(u_{n,k})\) and \(\widetilde{\Omega }_{n,k}:=\left\{ x\in \Omega _1: z_{n,k}^+(x)<\frac{2\,M}{\nu (\Omega _1)}\right\} \). Then,
from where
Case 1.1.1. Assume that \(\sup D(\gamma )=+\infty \). Let \(r_0\in {\mathbb {R}}\) be such that \(\gamma ^0(r_0)>2M/\nu (\Omega _1)\) and let \(k_0\in {\mathbb {N}}\) such that
Then, since in \(\widetilde{\Omega }_{n,k}\), \((\gamma _+)_k(u_{n,k})=z^+_{n,k}<\frac{2\,M}{\nu (\Omega _1)}\), from (2.26) we get that
Therefore, this bound, the generalised Poincaré-type inequality with \(l=\frac{\nu (\Omega _1)}{2}\) and (2.25) yield the boundedness of \(\{\Vert u_{n,k}^+\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\).
Case 1.1.2. Suppose that \(r_\gamma :=\sup D(\gamma )<+\infty \) and let \(h>0\). Since \(r_\gamma +h\not \in D(\gamma )\),
Take \(k_0\in {\mathbb {N}}\) such that \((\gamma _+)_k(r_\gamma +h)\ge \frac{2\,M}{\nu (\Omega _1)}\) for every \(k\ge k_0\). Then,
thus
Therefore, again, this bound together with the generalised Poincaré-type inequality with \(l=\frac{\nu (\Omega _1)}{2}\) and (2.22) yield the boundedness of \(\{\Vert u_{n,k}^+\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\).
Case 1.2. If \( \sup \text{ Ran }(\beta )=+\infty \), we proceed similarly.
Step 2. Using that \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) we obtain that \(\{\Vert u_{n,k}^-\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) is bounded with an analogous argument.
Consequently, we get that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) is bounded as desired.
Step B (Taking limits in n). The monotonicity properties obtained in Subsect. 2.2.2 together with the boundedness of \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) allow us to apply the monotone convergence theorem to obtain \(u_k\in L^p(\Omega ,\nu )\), \(k\in {\mathbb {N}}\), and \(u\in L^p(\Omega ,\nu )\) such that, taking a subsequence if necessary, \(u_{n,k}{\mathop {\rightarrow }\limits ^{n}} u_k\) in \(L^p(\Omega ,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega \) for \(k\in {\mathbb {N}}\), and \(u_{k}{\mathop {\rightarrow }\limits ^{k}} u\) in \(L^p(\Omega ,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega \).
We now want to take limits, in n and then in k, in (2.20) and (2.21). Since \(u_{n,k}{\mathop {\rightarrow }\limits ^{n}} u_k\) in \(L^p(\Omega ,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega \),
and
in \(L^{p'}(\Omega ,\nu )\) and, up to a subsequence, for \(\nu \)-a.e. \(x\in \Omega \). Indeed, the second and third limits follow because \(|u_{n,k}|^{p-2}u_{n,k}^+{\mathop {\rightarrow }\limits ^{n}}|u_{k}|^{p-2}u_{k}^+\) in \(L^{p'}(\Omega ,\nu )\). Now, since \(\{u_{n,k}\}_n\) is nonincreasing in n, \(|u_{n,k}|\le \max \{|u_{1,k}|,|u_k|\}\) \(\nu \)-a.e. in \(\Omega \), for every n, \(k\in {\mathbb {N}}\), so Lemma 2.1 yields the convergence (2.27) in \(L^{p'}(\Omega ,\nu )\).
Now, isolating \((\gamma _+)_k(u_{n,k})+(\gamma _-)_n(u_{n,k})\) and \((\beta _+)_k(u_{n,k})+(\beta _-)_n(u_{n,k})\) in equations (2.20) and (2.21), respectively, and taking the positive parts, we get that
for \(x\in \Omega _1\), and
for \(x\in \Omega _2\). Therefore, since the right-hand sides of these equations converge in \(L^{p'}(\Omega _1,\nu )\) and \(L^{p'}(\Omega _2,\nu )\) (and also \(\nu \)-a.e. in \(\Omega _1\) and \(\Omega _2\)), respectively, there exist \(z^+_k\in L^{p'}(\Omega _1,\nu )\) and \(\omega ^+_k\in L^{p'}(\Omega _2,\nu )\) such that \((\gamma _+)_k(u_{n,k}){\mathop {\rightarrow }\limits ^{n}}z^+_k\) in \(L^{p'}(\Omega _1,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega _1\), and \((\beta _+)_k(u_{n,k}){\mathop {\rightarrow }\limits ^{n}}\omega ^+_k\) in \(L^{p'}(\Omega _2,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega _2\). Moreover, since \((\gamma _+)_k\) and \((\beta _+)_k\) are maximal monotone graphs, \(z^+_k= (\gamma _+)_k(u_{k})\) \(\nu \)-a.e. in \(\Omega _1\), and \(\omega ^+_k= (\beta _+)_k(u_{k})\) \(\nu \)-a.e. in \(\Omega _2\).
Similarly, taking the negative parts, there exist
and
Moreover, by [15, Lemma G], \(z^-_k\in \gamma _-(u_{k})\) and \(\omega ^-_k\in \beta _-(u_{k})\). Therefore, we have obtained that
for \(\nu \)-a.e. \(x\in \Omega _1\), and
for \(\nu \)-a.e. \(x\in \Omega _2\).
Step C (Taking limits in k). Now again, isolating \(z_k^++z^-_k\) and \(\omega _k^++\omega _k^-\) in equations (2.28) and (2.29), respectively, and taking the positive and negative parts as above, we get that there exist \(z^+\in L^{p'}(\Omega _1,\nu )\), \(z^-\in L^{p'}(\Omega _1,\nu )\), \(\omega ^+\in L^{p'}(\Omega _2,\nu )\) and \(\omega ^-\in L^{p'}(\Omega _2,\nu )\) such that \(z^+_k {\mathop {\rightarrow }\limits ^{k}}z^+\) and \(z^-_k {\mathop {\rightarrow }\limits ^{k}}z^-\) in \(L^{p'}(\Omega _1,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega _1\), and \(\omega ^+_k {\mathop {\rightarrow }\limits ^{k}}\omega ^+\) and \(\omega ^-_k {\mathop {\rightarrow }\limits ^{k}}\omega ^-\) in \(L^{p'}(\Omega _2,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega _2\). In addition, by the maximal monotonicity of \(\gamma _-\) and \(\beta _-\), \(z^-\in \gamma _-(u)\) and \(\omega ^-\in \beta _-(u)\) \(\nu \)-a.e. in \(\Omega _1\) and \(\Omega _2\), respectively. Moreover, by [15, Lemma G], \(z^+\in \gamma _+(u)\) and \(\omega ^+\in \beta _+(u)\) \(\nu \)-a.e. in \(\Omega _1\) and \(\Omega _2\), respectively.
Consequently,
and
where \(z=z^++z^- \in \gamma (u)\) \(\nu \)-a.e. in \(\Omega _1\) and \(\omega =\omega ^++\omega ^-\in \beta (u)\) \(\nu \)-a.e. in \(\Omega _2\). The proof of existence in this case is done. \(\square \)
Proof
(Proof of Theorem 2.7 when \(\mathcal {R}_{\gamma ,\beta }^\pm \) are finite) Suppose that
Let \(\varphi \in L^{p'}(\Omega ,\nu )\), and assume that it satisfies
Then, for \(\varphi _{n,k}\) defined as in (2.19), there exist \(M_1, M_2\in {\mathbb {R}}\) and \(n_0, k_0\in {\mathbb {N}}\) such that
for every \(n\ge n_0\) and \(k\ge k_0\). For \(n, k\in {\mathbb {N}}\) let \(u_{n,k}\in L^\infty (\Omega ,\nu )\) be the solution of the Approximate Problem (2.20)–(2.21), and let
Observe that \(M_3\) is finite by the generalised Poincaré-type inequality together with (2.25). Let \(k_1\in {\mathbb {N}}\) such that \(k_1\ge k_0\) and \(M_1+\frac{1}{k}M_3\nu (\Omega )^{\frac{1}{p(p-1)}}<\mathcal {R}_{\gamma ,\beta }^+\) for every \(k\ge k_1\).
Step D (Boundedness of \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_n\) and passing to the limit in n) Let us see that, for each \(k\in \mathbb {N}\), \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_n\) is bounded. Fix \(k\ge k_1\) and suppose that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is not bounded. Then, by (2.31), since \(u_{n,k}\) is nondecreasing in n,
Thus, using again that \(u_{n,k}\) is nondecreasing in n, there exists \(n_1\ge n_0\) such that
for every \(n\ge n_1\), and thus
Consequently, \(\Vert u_{n,k}^-\Vert _{L^{p-1}(\Omega ,\nu )}\le M_3\nu (\Omega )^{\frac{1}{p(p-1)}}\) for \(n\ge n_1\). Then, with this bound and (2.30) at hand, integrating (2.20) and (2.21) with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, adding both equations and neglecting some nonnegative terms we get
Therefore, for each \(n\in {\mathbb {N}}\), either
or
where \(\delta :=\mathcal {R}_{\gamma ,\beta }^+- M_4>0\).
For \(n\in {\mathbb {N}}\) such that (2.32) holds let
Then
and
Therefore,
thus \(\nu (K_{n,k})>0\), \(\displaystyle \sup \text{ Ran }(\gamma )-\inf \text{ Ran }(\gamma )-\frac{\delta }{4\nu (\Omega _1)}>0\) and
Note that, if \(\sup \text{ Ran }(\gamma )-\frac{\delta }{4\nu (\Omega _1)}\le 0\), then \(z_{n,k}\le 0\) in \(K_{n,k}\), thus \(u_{n,k}^+=0\) in \(K_{n,k}\) and, consequently, \(\Vert u_{n,k}^+ \Vert _{L^p(K_{n,k},\nu )}=0\). Therefore, by the generalised Poincaré-type inequality and (2.25) we get that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is bounded, which is a contradiction. We may therefore suppose that \(\sup \text{ Ran }(\gamma ) -\frac{\delta }{4\nu (\Omega _1)} > 0\). Then, for \(k_2\ge k_1\) large enough so that \(\sup \text{ Ran }((\gamma _+)_{k})>\sup \text{ Ran }(\gamma )-\frac{\delta }{4\nu (\Omega _1)}\) for \(k\ge k_2\),
and by the generalised Poincaré-type inequality and (2.25) we get that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is bounded, which is a contradiction. Similarly, for \(n\in {\mathbb {N}}\) such that (2.33) holds.
We have obtained that \(\{\Vert u_{n,k}\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is bounded for each \(k\in {\mathbb {N}}\). Therefore, since \(\{u_{n,k}\}_n\) is nondecreasing in n, we may apply the monotone convergence theorem to obtain \(u_k\in L^p(\Omega ,\nu )\), \(k\in {\mathbb {N}}\), such that \(u_{n,k}{\mathop {\rightarrow }\limits ^{n}} u_k\) in \(L^p(\Omega ,\nu )\) and pointwise \(\nu \)-a.e. in \(\Omega \) for \(k\in {\mathbb {N}}\). Proceeding now like in Step B of the previous proof we get: \(z_k^+\in L^{p'}(\Omega _1,\nu )\) and \(\omega _k^+\in L^{p'}(\Omega _2,\nu )\) such that \(z^+_k\in \gamma _+(u_{k})\) and \(\omega ^+_k\in \beta _+(u_{k})\) \(\nu \)-a.e. in \(\Omega _1\) and \(\Omega _2\), respectively; and \(z_k^-\in L^{p'}(\Omega _1,\nu )\) and \(\omega _k^-\in L^{p'}(\Omega _2,\nu )\) with \(z^-_k\in \gamma _-(u_{k})\) and \(\omega ^-_k\in \beta _-(u_{k})\), \(\nu \)-a.e. \(\Omega _1\) and \(\Omega _2\), respectively, and such that
for \(\nu \)-a.e. every \(x\in \Omega _1\), and
for \(\nu \)-a.e. every \(x\in \Omega _2\).
Step E (Boundedness of \(\{\Vert u_{k}\Vert _{L^p(\Omega ,\nu )}\}_k\) and passing to the limit in k) We now see that \(\{\Vert u_{k}\Vert _{L^p(\Omega ,\nu )}\}_{k}\) is bounded. Since \(u_k^+\le u_1^+\), it is enough to see that \(\{\Vert u_{k}^-\Vert _{L^p(\Omega ,\nu )}\}_{k}\) is bounded.
Therefore, for each \(k\in {\mathbb {N}}\), either
or
where \(\delta ':=M_2-\mathcal {R}_{\gamma ,\beta }^->0\).
For \(k\in {\mathbb {N}}\) such that (2.36) holds let \(K_{k}:=\{x\in \Omega _1 \,: \, z_{k}(x)>\inf \text{ Ran }(\gamma ) +\frac{\delta '}{4\nu (\Omega _1)}\}\). Then,
and
Therefore,
thus \(\nu (K_{k})>0\), \(\displaystyle \sup \text{ Ran }(\gamma ) -\inf \text{ Ran }(\gamma ) -\frac{\delta '}{4\nu (\Omega _1)}>0\) and
Now, if \(\inf \text{ Ran }(\gamma ) +\frac{\delta '}{4\nu (\Omega _1)}\ge 0\) then \(z_{k}\ge 0\) in \(K_{k}\), thus \(u_{n,k}^-=0\) in \(K_{k}\) and \(\Vert u_{k}^-\Vert _{L^p(K_{k},\nu )}=0\); so by the generalised Poincaré-type inequality and (2.25) we get that \(\{\Vert u_{k}\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is bounded. If \(\inf \text{ Ran }(\gamma )+\frac{\delta '}{4\nu (\Omega _1)} < 0\), then
and by the generalised Poincaré inequality and (2.25) we get that \(\{\Vert u_{k}\Vert _{L^p(\Omega ,\nu )}\}_{k}\) is bounded. Similarly, for \(k\in {\mathbb {N}}\) such that (2.37) holds.
Now, proceeding as in Step C of the previous proof, we finish this proof. \(\square \)
Finally, we give the proof of the remaining case.
Proof
(Proof of Theorem 2.7 in the mixed case) Let us see the existence for
or
Suppose that (2.38) holds and let \(\varphi \in L^{p'}(\Omega ,\nu )\) satisfying
If (2.39) holds and we have \(\varphi \in L^{p'}(\Omega ,\nu )\) satisfying \(\displaystyle \int _{\Omega }\varphi {\text {d}}\nu < \mathcal {R}_{\gamma ,\beta }^+\), the argument is analogous.
Let \(\varphi _{n,k}\) be defined as in (2.19) and let \(u_{n,k}\in L^\infty (\Omega ,\nu )\), \(n, k\in {\mathbb {N}}\), be the solution of the Approximate Problem (2.20)–(2.21). Then, by Lemma A.7 together with (2.22), \(\{\Vert u_{n,k}^+\Vert _{L^p(\Omega ,\nu )}\}_{n,k}\) is bounded. However, for a fixed \(k\in {\mathbb {N}}\), since \(u_{n,k}\) is nondecreasing in n, \(\{\Vert u_{n,k}^-\Vert _{L^p(\Omega ,\nu )}\}_{n}\) is also bounded. Therefore, proceeding as in Step B of the first case, we obtain \(u_k\in L^p(\Omega ,\nu )\), \(z_k^+\), \(z_k^-\in L^{p'}(\Omega _1,\nu )\) and \(\omega _k^+\), \(\omega _k^-\in L^{p'}(\Omega _2,\nu )\), \(k\in {\mathbb {N}}\), such that
for \(\nu \)-a.e. \(x\in \Omega _1\), and
for \(\nu \)-a.e. \(x\in \Omega _2\); where, for \(k\in {\mathbb {N}}\),
and
We now prove that \(\{\Vert u_{k}\Vert _{L^p(\Omega ,\nu )}\}_{k}\) is bounded. Proceeding as in Step E of the previous proof and using the same notation, we get that for each \(k\in {\mathbb {N}}\), either
or
Case 1. For \(k\in {\mathbb {N}}\) such that (2.41) holds, let
Then,
Now, by (2.40),
Thus, for a constant \(D_1\) independent of k and h,
Hence, by (2.4) and (2.25), there exist constants \(D_2\) and \(D_3\), independent of k and h, such that
Consequently, we may find \(h>0\) such that
Therefore,
Recalling (2.43), we get
thus
Consequently, \(\displaystyle h-\inf \text{ Ran }(\gamma ) -\frac{\delta '}{4\nu (\Omega _1)}>0\) and
From here we conclude as in the previous proof.
Case 2. For \(k\in {\mathbb {N}}\) such that (2.42) holds, let
and proceed similarly. \(\square \)
Remark 2.8
-
(i)
Taking limits in (2.25) we obtain that, if [u, v] is a solution of \((GP_\varphi ^{\textbf{a}_p,\gamma ,\beta })\), then
$$\begin{aligned} \displaystyle{} & {} \frac{c_p}{2}\left( \int _{\Omega }\int _{\Omega }|u(y)-u(x)|^p dm_x(y){\text {d}}\nu (x)\right) ^{\frac{1}{p'}} \displaystyle \\{} & {} \le \Lambda _1\Vert \varphi \Vert _{L^{p'}(\Omega ,\nu )}+ \frac{\Lambda _1+\Lambda _2}{\nu (\Omega )^{\frac{1}{p}} }\Vert \varphi \Vert _{L^{1}(\Omega ,\nu )}, \end{aligned}$$where \(c_p\) is the constant in (2.5), and \(\Lambda _1\) and \(\Lambda _2\) come from the generalised Poincaré-type inequality and depend only on p, \(\Omega _1\) and \(\Omega _2\).
-
(ii)
Observe that, on account of (2.4) and the above estimate, we have
$$\begin{aligned}{} & {} \left( \int _{\Omega }\left| \int _{\Omega }\textbf{a}_p(x,y,u(y)-u(x))dm_x(y)\right| ^{p'}{\text {d}}\nu (x)\right) ^{\frac{1}{p'}} \\{} & {} \le C_p\nu (\Omega )+ \frac{2C_p}{c_p}(2\Lambda _1+\Lambda _2)\Vert \varphi \Vert _{L^{p'}(\Omega ,\nu )}. \end{aligned}$$Therefore, since [u, v] is a solution of \((GP_\varphi ^{\textbf{a}_p,\gamma ,\beta })\),
$$\begin{aligned} \begin{array}{l} \displaystyle \Vert v\Vert _{L^{p'}(\Omega ,\nu )} \le C_p\nu (\Omega )+\left( \frac{2C_p}{c_p}(2\Lambda _1+\Lambda _2)+1\right) \Vert \varphi \Vert _{L^{p'}(\Omega ,\nu )}. \end{array} \end{aligned}$$ -
(iii)
When \(\varphi =0\) in \(\Omega _2\), we can easily get that \(v\ll \varphi \) in \(\Omega _1.\)
2.3 Other boundary conditions
We can now ask for existence and uniqueness of solutions of the following problem (which is introduced in Sect. 2.1)
or, of the more general problem,
Recall that \(\mathcal {N}^{\textbf{a}_p}_2\) is defined as follows:
which involves integration with respect to \(\nu \) only over W, or more specifically over \(\partial _m(X\setminus W)\).
For Problem (2.44), we know that, in general, we do not have an appropriate Poincaré-type inequality to work with (see Remark A.5). Therefore, other techniques must be used to obtain the existence of solutions. In the particular case of \(\gamma (r)=\beta (r)=r\), this was done in [43] by exploiting further monotonicity techniques.
However, if a generalised Poincaré-type inequality (as defined in Definition A.1) is satisfied on \((A,B)=(\Omega _1, \Omega _2)\), we could solve the above problem by using the same techniques that we have used to solve Problem (2.7). Indeed, we can work analogously but with the integration by parts formula given in Remark 2.9. Note that this kind of Poincaré-type inequality holds, for example, for finite graphs; even if \(\Omega _2=\partial _m W\).
Remark 2.9
Let \(\Omega :=\Omega _1\cup \Omega _2\). The following integration by parts formula holds: Let u be a measurable function such that
and let \(w \in L^{q'}(\Omega ,\nu )\). Then,
Remark 2.10
It is possible to consider this type of problems but with the random walk and the nonlocal Leray–Lions operator having a different behaviour on each subset \(\Omega _i\), \(i=1, 2\). For example, one could consider a problem, posed in \(\Omega _1\cup \Omega _2\subset \mathbb {R}^N\), such as the following:
where \(J_i\) are kernels like the one in Example 1.2, and \(\textbf{a}_p^i\) are functions like the one in Subsect. 2.1, \(i=1,2,3\). This could be done by obtaining a Poincaré-type inequality involving \(\frac{1}{\alpha _0}J_0\), where \(J_0\) is the minimum of the previous three kernels and \(\alpha _0=\int _{\mathbb {R}^N}J_0(z)dz\). This idea has been used in [22] to study a homogenization problem.
3 Doubly nonlinear diffusion problems
We study two kinds of nonlocal p-Laplacian-type diffusions problems. In one of them, we cover nonlocal nonlinear diffusion problems with nonlinear dynamical boundary condition; and on the other, we tackle nonlinear boundary conditions. We work under Assumptions 1–5 used in Subsect. 2.2.
3.1 Nonlinear dynamical boundary conditions
Our aim in this subsection is to study the following diffusion problem:
of which Problem (1.2) is a particular case and which covers the case of dynamic evolution on the boundary \(\partial _mW\) when \(\beta \ne \mathbb {R}\times \{0\}\). This includes, in particular, for \(\gamma =\mathbb {R}\times \{0\}\), the problem where the dynamic evolution occurs only on the boundary:
See [4] for the reference local model.
Note that we may abbreviate Problem (3.1) by using v instead of (v, w) and f instead of (f, g) as
To solve this problem, we use nonlinear semigroup theory. To this end, we introduce a multivalued operator associated with Problem (3.2) that allows us to rewrite it as an abstract Cauchy problem. Observe that this operator is defined on \(L^1(\Omega ,\nu ) \equiv \left( L^1(\Omega _1,\nu )\times L^1(\Omega _2,\nu )\right) .\)
Definition 3.1
We say that \( (v, {\hat{v}}) \in \mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p}\) if \(v,{\hat{v}} \in L^1(\Omega ,\nu )\), and there exists \( u\in L^p(\Omega ,\nu )\) with
and
such that
and
that is, [u, v] is a solution of \((GP_{v+{{\hat{v}}}})\) (see (2.8) and Definition 2.4).
On account of the results given in Subsect. 2.2 (Theorems 2.6 and 2.7), we have the following result. Recall that an operator A in \(L^1(\Omega ,\nu )\) is T-accretive if
In fact, A is T-accretive if, and only if, its resolvents are contractions and order-preserving (see, for example, [8, Appendix] for further details).
Theorem 3.2
The operator \(\mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p}\) is T-accretive in \(L^1(\Omega ,\nu )\) and satisfies the range condition
With respect to the domain of such operator, we can prove the following result.
Theorem 3.3
It holds that
Therefore, we also have that
Proof
It is obvious that
For the other inclusion, it is enough to see that
Suppose first that \(\gamma \) and \(\beta \) satisfy
It is enough to see that for any \(v\in L^\infty (\Omega ,\nu )\) such that there exist \(m_1<0\), \(\widetilde{m_i}\in {\mathbb {R}}\), \(\widetilde{M_i}\in {\mathbb {R}}\), \(M_i>0\), \(i=1,2\), satisfying
it holds that \(v\in \overline{D(\mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p})}^{L^{p'}(\Omega ,\nu )}\).
By the results in Subsect. 2.2.4, we know that for \(n\in {\mathbb {N}}\), there exists \(u_n\in L^p(\Omega ,\nu )\) and \(v_n\in L^{p'}(\Omega ,\nu )\) such that \([u_n,v_n]\) is a solution of \(\displaystyle \left( GP_v^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \), i.e., \(v_n\in \gamma (u_n)\) \(\nu \)-a.e. in \(\Omega _1\), \(v_n\in \beta (u_n)\) \(\nu \)-a.e. in \(\Omega _2\) and
In other words, \((v_n,n(v-v_n))\in \mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p}\) or, equivalently,
Let us see that \(v_n{\mathop {\longrightarrow }\limits ^{n}} v\) in \(L^{p'}(\Omega ,\nu )\).
Let \(a_{m_1}\le 0\) and \(a_{M_1}\ge 0\) such that
and let \(b_{M_2}\ge 0\) such that
Set
and
Then, \([\widehat{u},\widehat{v}]\) is a solution of \(\displaystyle \left( GP_{\widehat{\varphi }_n}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \).
Similarly, for
and
we have that \([\widetilde{u},\widetilde{v}]\) is a solution of \(\displaystyle \left( GP_{\widetilde{\varphi }_n}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \).
Now, recalling (2.4), we have that there exists \(n_0\in {\mathbb {N}}\) such that
and
for \(n\ge n_0\). Consequently, by the maximum principle (Theorem 2.6), we obtain that
thus
Finally, since
we conclude that, on account of (2.4),
The other cases follow similarly, we see two of them. Note that, since \(\mathcal {R}_{\gamma ,\beta }^-<\mathcal {R}_{\gamma ,\beta }^+\), it is not possible to have \(\gamma =\mathbb {R}\times \{0\}\) and \(\beta =\mathbb {R}\times \{0\}\) simultaneously. For example, suppose that we have
We use the same notation. Let \(v\in L^\infty (\Omega ,\nu )\) such that there exist \(\widetilde{m_i}\in {\mathbb {R}}\), \(\widetilde{M_i}\in {\mathbb {R}}\), \(M_i>0\), \(i=1,2\), satisfying
As before, the results in Subsect. 2.2.4 ensure that there exist \(u_n\in L^p(\Omega ,\nu )\) and \(v_n\in L^{p'}(\Omega ,\nu )\), \(n\in {\mathbb {N}}\), such that \([u_n,v_n]\) is a solution of \(\displaystyle \left( GP_v^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Let \(\ a_{M_1}\ge 0\) and \(b_{M_2}\ge 0\) such that
Now again, let
and
Then, as before, \([\widehat{u},\widehat{v}]\) is a solution of \(\displaystyle \left( GP_{\widehat{\varphi }_n}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \).
Now, taking \(\widetilde{v}\), \(\widetilde{u}\) and \({\widetilde{\varphi }}\) all equal to the null function in \(\Omega \) and recalling that \(\textbf{a}_p(x,y,0)=0\) for every \(x, y\in X\), we obviously have that \([\widetilde{u},\widetilde{v}]\) is a solution of \(\displaystyle \left( GP_{0}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Consequently, again by the second part of the maximum principle, we obtain, as desired, that \(0\le u_n\le \widehat{v}\) for n large enough.
Finally, as a further example of a case which does not follow exactly with the same argument, suppose that \(\gamma :={\mathbb {R}}\times \{0\}\) and, for example,
In this case, we take \(0\not \equiv v\in L^\infty (\Omega ,\nu )\) such that \(v=0\) in \(\Omega _1\) and
As in the previous cases, there exist \(u_n\in L^p(\Omega ,\nu )\) and \(v_n\in L^{p'}(\Omega ,\nu )\), \(n\in {\mathbb {N}}\), such that \([u_n,v_n]\) is a solution of \(\displaystyle \left( GP_v^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Let \(b_{M_2}\ge 0 \) such that \(M_2\in \beta (b_{M_2})\),
and
Then, \([\widehat{u},\widehat{v}]\) is a solution of \(\displaystyle \left( GP_{\varphi _n}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Finally, take \(\widetilde{v}\) and \(\widetilde{u}\) again equal to the null function in \(\Omega \) so that \([\widetilde{u},\widetilde{v}]\) is a solution of \(\displaystyle \left( GP_{0}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Consequently, for n large enough, we get that \(0\le u_n\le \widehat{v}\). \(\square \)
In the next result, we state the existence and uniqueness of solutions of Problem (3.2).
Theorem 3.4
Let \(T>0\). For any \(v_0\in L^{1}(\Omega ,\nu )\) and \(f\in L^1(0,T;L^{1}(\Omega ,\nu ))\) such that
and
there exists a unique mild-solution \(v\in C([0,T];L^1(\Omega ,\nu ))\) of Problem (3.2).
Let v and \({\widetilde{v}}\) be the mild solutions of Problem (3.2) with respective data \(v_0,\ {\widetilde{v}}_0\in L^{1}(\Omega ,\nu )\) and \(f,\ {\widetilde{f}}\in L^1(0,T;L^{1}(\Omega ,\nu ))\). Then,
If, in addition to the previous assumptions on the data, we impose that
then the mild solution v belongs to \(W^{1,1}(0,T;L^{1}(\Omega ,\nu ))\) and satisfies
that is, v is a strong solution.
Proof
We start by proving the existence of mild solutions. For \(n\in {\mathbb {N}}\), consider the partition
where \(t_i^n:=iT/n\), \(i=1,\ldots ,n\). Given \(\epsilon >0\), there exists \(n\in \mathbb {N}\), \(f_i^n\in L^{p'}(\Omega ,\nu )\), \(i=1,\ldots n\), and \(v_0^n \in \overline{D(\mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p})}^{L^{p'}(\Omega ,\nu )}\) (i.e., \(v_0^n\in L^{p'}(\Omega ,\nu )\) satisfying \(\varGamma ^-\le v_0^n\le \varGamma ^+\) \(\nu \)-a.e. in \(\Omega _1\), and \(\mathfrak {B}^-\le v_0^n\le \mathfrak {B}^+\) \(\nu \)-a.e. in \(\Omega _2\)) such that \(T/n\le \epsilon \),
and
Then, setting
we have that
By the results in Subsect. 2.2.4 we see that, for n large enough, we may recursively find a solution \([u_i^n,v_i^n]\) of \(\displaystyle \left( GP^{\frac{T}{n}\textbf{a}_p,\gamma ,\beta }_{\frac{T}{n} f_i^n+v_{i-1}^n}\right) \), \(i=1,\ldots ,n\), in other words,
or, equivalently,
with \(v_i^n(x)\in \gamma (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _1\) and \(v_i^n(x)\in \beta (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _2\), \(i=1,\ldots ,n\). That is, we may find the unique solution \(v_i^n\) of the time discretization scheme associated with (3.2):
However, to apply the results in Subsect. 2.2.4, we must ensure that
holds for each step. For the first step we need that
holds so that condition (3.8) is satisfied. Integrating (3.7) with respect to \(\nu \) over \(\Omega \), we get
thus
so that, for the second step, we need
Therefore, we recursively obtain that for each n and each step \(i=1,\ldots , n\), the following must be satisfied:
However, taking n large enough, this holds thanks to (3.3), (3.5) and (3.6).
Therefore,
is an \(\epsilon \)-approximate solution of Problem (3.2) as defined in nonlinear semigroup theory. Consequently, by nonlinear semigroup theory (see [11, 10, Theorem 4.1], or [8, Theorem A.27]) and on account of Theorem 3.2 and Theorem 3.3 we have that Problem (3.2) has a unique mild solution \(v\in C([0,T];L^1(\Omega ,\nu ))\) with
Uniqueness and the maximum principle for mild solutions are guaranteed by the T-accretivity of the operator.
Let us now see that v is a strong solution of Problem (3.2) when (3.4) holds. Note that, since \(v_0\in L^{p'}(\Omega ,\nu )\), we may take \(v_0^n=v_0\) for every \(n\in {\mathbb {N}}\) in the previous computations and \(f_i^n\in L^{p'}(\Omega ,\nu )\), \(i=1,\ldots n\), additionally satisfying
Let us define
Multiplying equation (3.7) by \(u_i^n\) and integrating over \(\Omega \) with respect to \(\nu \), we obtain
Now, since \(v_i^n(x)\in \gamma (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _1\) and \(v_i^n(x)\in \beta (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _2\),
Consequently,
Therefore, from (3.10), it follows that
\(i=1,\ldots ,n\). Then, integrating this equation over \(]t_{i-1}^n,t_i^n]\) and adding for \(1\le i \le n\), we get
which, recalling the definitions of \(f_n\), \(u_n\) and \(v_n\), and integrating by parts, can be rewritten as:
This, together with (2.5) and the fact that \(j^*_\gamma \) and \(j^*_\beta \) are nonnegative, yields
Therefore, for any \(\delta >0\), by (3.4) and Young’s inequality, there exists \(C(\delta )>0\) such that
Now, by (3.9), if \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \), there exists \(M>0\) and \(n_0\in {\mathbb {N}}\) such that
and, if \(\mathcal {R}_{\gamma ,\beta }^+<+\infty \), there exist \(M\in {\mathbb {R}}\), \(h>0\) and \(n_0\in {\mathbb {N}}\) such that
and
Consequently, Lemma A.7 and Lemma A.8 yield
and for some constant \(C_2>0\). Similarly, we may find \(C_3>0\) such that
Consequently, by (3.12), choosing \(\delta \) small enough, we deduce that \(\{u_{n}\}_n\) is bounded in \(L^p(0,T;L^p(\Omega ,\nu ))\). Therefore, there exists a subsequence, which we continue to denote by \(\{u_{n}\}_n\), and \(u\in L^p(0,T;L^p(\Omega ,\nu ))\) such that
Then, since \(\gamma \) and \(\beta \) are maximal monotone graphs, we conclude that \(v(t)(x)\in \gamma (u(t)(x))\) for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in (0,T)\times \Omega _1\) and \(v(t)(x)\in \beta (u(t)(x))\) for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in (0,T)\times \Omega _2\).
Note that, since, by (3.12),
then, by (2.4), \(\{[(t,x,y)\mapsto \textbf{a}_p(x,y,u_{n}(t)(y)-u_{n}(t)(x))]\}_n\) is bounded in \(L^{p'}(0,T; L^{p'}(\Omega \times \Omega ,\nu \otimes m_x))\) so we may take a further subsequence, which we still denote in the same way, such that
Note that, for any \(\xi \in L^p(\Omega ,\nu )\), by the integrations by parts formula we know that
for \(t\in [0,T]\), thus taking limits as \(n\rightarrow \infty \) we have
Now, from (3.7), we have that
for \(t\in [0,T]\) and \(x\in \Omega \). Let \(\Psi \in W^{1,1}_0(0,T;L^p(\Omega ,\nu ))\), \(\hbox {supp}(\Psi )\subset \subset [0,T]\), then
for \(x\in \Omega \). Therefore, multiplying (3.14) (for the previously chosen subsequence) by \(\Psi \), integrating over \((0,T)\times \Omega \) with respect to \(\mathcal {L}^1\otimes \nu \) and taking limits, we get
Therefore, taking \(\Psi (t)(x)=\psi (t)\xi (x)\), where \(\psi \in C_c^\infty (0,T)\) and \(\xi \in L^p(\Omega ,\nu )\), we obtain that
It follows that
Therefore, since \(v\in C([0,T];L^1(\Omega ,\nu ))\), \(\Phi \in L^{p'}(0,T; L^{p'}(\Omega \times \Omega ,\nu \otimes m_x))\) and \(f\in L^{p'}(0,T;L^{p'}(\Omega ,\nu ))\), we have \(v'\in L^{p'}(0,T;L^{p'}(\Omega ,\nu ))\) and \(v\in W^{1,1}(0,T; L^{1}(\Omega ,\nu ))\).
Hence, to conclude it remains to prove that
for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in [0,T]\times \Omega \). To this aim, we make use of the following claim that will be proved later on:
Now, let \(\rho \in L^p(0,T;L^p(\Omega ,\nu ))\). By (2.3), we have
Thus, taking limits as \(n\rightarrow \infty \) and using (3.16), we obtain
which, integrating by parts and recalling (3.13), becomes
To conclude, take \(\rho =u\pm \lambda \xi \) for \(\lambda >0\) and \(\xi \in L^p(0,T;L^p(\Omega ,\nu ))\) to get
which, letting \(\lambda \rightarrow 0 \) yields
for any \(\xi \in L^p(0,T;L^p(\Omega ,\nu ))\). Therefore,
for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in [0,T]\times \Omega \).
Let us prove claim (3.16). By (3.11) and Fatou’s lemma, we have
Moreover, by (3.15),
where F is given by
Let \(\psi \in C_c^\infty (0,T)\), \(\psi \ge 0\), \(\tau >0\) and
Then, for \(\tau \) small enough, \(\eta _\tau \in W^{1,1}_0(0,T;L^p(\Omega ,\nu ))\) and we may use it as a test function in (3.18) to obtain
Now,
thus, for \(v,{\hat{v}} \in \gamma (u)\),
A similar fact holds for \(\beta \). Then, for \(\tau >0\) fixed, since \(v(t)(x)\in \gamma (u(t)(x))\) for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in (0,T)\times \Omega _1\) and \(v(t)(x)\in \beta (u(t)(x))\) for \(\mathcal {L}^1\otimes \nu \)-a.e. \((t,x)\in (0,T)\times \Omega _2\),
Letting \(\tau \rightarrow 0^+\) in the above expression, by the dominated convergence theorem,
Taking
yields the opposite inequality so that, in fact,
Then,
in \(\mathcal {D}'(]0,T[)\), thus, in particular,
Therefore, integrating from 0 to T in (3.20) and recalling (3.19), we get
which, together with (3.17), yields the claim (3.16). \(\square \)
Observe that we have imposed the compatibility condition (3.3) because, for a strong solution,
Example 3.5
Let \(W\subset X\) be a measurable set such that \(W_m\) is m-connected. Given \(f\in L^1(\partial _mW,\nu )\), we say that a function \(u\in L^1(W_m,\nu )\) is an \(\textbf{a}_p\)-lifting of f to \(W_m=W\cup \partial _mW\) if
We define the Dirichlet-to-Neumann operator \(\mathfrak {D}_{\textbf{a}_p}\subset L^1(\partial _mW,\nu )\times L^1(\partial _mW,\nu )\) as follows: \((f,\psi )\in \mathfrak {D}_{\textbf{a}_p}\) if
where u is an \(\textbf{a}_p\)-lifting of f to \(W_m\).
Then, rewriting the operator \(\mathfrak {D}_{\textbf{a}_p}\) as \(\mathcal {B}^{m,\gamma ,\beta }_{\textbf{a}_p}\) for \(\gamma (r)=0\) and \(\beta (r)=r\), \(r\in {\mathbb {R}}\), (\(\Omega _1=W\) and \(\Omega _2=\partial _mW\)), by the results in this subsection we have that \(\mathfrak {D}_{\textbf{a}_p}\) is T-accretive in \(L^1(\partial _mW,\nu )\) (it is easy to see that, in fact, in this situation, it is completely accretive), it satisfies the range condition
and it has dense domain. The non-homogeneous Cauchy evolution problem for this nonlocal Dirichlet-to-Neumann operator is a particular case of Problem (3.2):
See, for example, [2, 3, 24, 37, 44] and the references therein, for other evolution problems with p-Dirichlet-to-Neumann operators, see [16] for the problem with convolution kernels.
3.2 Nonlinear boundary conditions
In this subsection, our aim is to study the following diffusion problem:
that in particular covers Problem (1.1). See [15] for the reference local model.
We assume that
since, otherwise, we do not have an evolution problem. Hence, \(\mathcal {R}_{\gamma ,\beta }^-<\mathcal {R}_{\gamma ,\beta }^+.\) Moreover, we also assume that
since the case \( \mathfrak {B}^-=\mathfrak {B}^+\) (\(\beta ={\mathbb {R}}\times \{0\}\)) is treated with more generality in Subsect. 3.1.
We will again make use of nonlinear semigroup theory. To this end, we introduce the corresponding operator associated with \(\left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \), which is now defined in \(L^1(\Omega _1,\nu )\).
Definition 3.6
We say that \((v,{\hat{v}}) \in B^{m,\gamma ,\beta }_{\textbf{a}_p}\) if \( v,{\hat{v}} \in L^1(\Omega _1,\nu )\) and there exist \( u\in L^p(\Omega ,\nu )\) and \(w\in L^1(\Omega _2,\nu )\) with
and
such that
and
that is, [u, (v, w)] is a solution of \((GP_{(v+{\hat{v}},\textbf{0})})\), where \(\textbf{0}\) is the null function in \(\Omega _2\) (see (2.8) and Definition 2.4).
Set
On account of the results given in Subsect. 2.2 (Theorems 2.6 and 2.7), we have:
Theorem 3.7
The operator \(B^{m,\gamma ,\beta }_{\textbf{a}_p}\) is T-accretive in \(L^1(\Omega ,\nu )\) and satisfies the range condition
Remark 3.8
Observe that if \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \), then the closure of \(B^{m,\gamma ,\beta }_{\textbf{a}_p}\) is m-T-accretive in \(L^1(\Omega _1,\nu )\).
With respect to the domain of this operator, we prove the following result.
Theorem 3.9
Therefore, we also have
Proof
It is obvious that
For the other inclusion, it is enough to see that
We work on a case-by-case basis.
(A) Suppose that \(\varGamma ^-<0<\varGamma ^+ \). It is enough to see that for any \(v\in L^\infty (\Omega _1,\nu )\) such that there exist \(m\in {\mathbb {R}}\), \(\widetilde{m}<0\), \(\widetilde{M}>0\), \(M\in {\mathbb {R}}\) satisfying
it holds that \(v\in \overline{D(B^{m,\gamma ,\beta }_{\textbf{a}_p})}^{L^{p'}(\Omega _1,\nu )}\).
By the results in Subsect. 2.2.4 we know that, for \(n\in {\mathbb {N}}\), there exist \(u_n\in L^p(\Omega ,\nu )\), \(v_n\in L^{p'}(\Omega _1,\nu )\) and \(w_n\in L^{p'}(\Omega _2,\nu )\), such that \([u_n,(v_n,\frac{1}{n} w_n)]\) is a solution of \(\left( GP_{(v,\textbf{0})}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \), i.e., \(v_n\in \gamma (u_n)\) \(\nu \)-a.e. in \(\Omega _1\), \(w_n\in \beta (u_n)\) \(\nu \)-a.e. in \(\Omega _2\) and
In other words, \((v_n,n(v-v_n))\in B^{m,\gamma ,\beta }_{\textbf{a}_p}\) or, equivalently,
Let us see that \(v_n{\mathop {\longrightarrow }\limits ^{n}} v\) in \(L^{p'}(\Omega _1,\nu )\).
(A1) Suppose first that \(\hbox {sup} D(\beta )=+\infty \). Take \(a_M>0\) such that \(M\in \gamma (a_M)\) and let \(N\in \beta (a_M)\). Let
and
Then, \([\widehat{u}, \widehat{v}]\) is a supersolution of \(\big (GP^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }_{\varphi }\big )\) and \((v,\textbf{0})\le \varphi \). Thus, by the maximum principle (Theorem 2.6),
(A2) Suppose now that \(\hbox {sup}D(\beta )=r_\beta <+\infty \). Again, by the results in Subsect. 2.2.4 we know that, for \(n\in {\mathbb {N}}\), there exist \(\widetilde{u}_n\in L^p(\Omega ,\nu )\), \(\widetilde{v}_n\in L^{p'}(\Omega _1,\nu )\) and \(\widetilde{w}_n\in L^{p'}(\Omega _2,\nu )\), such that \([\widetilde{u}_n,(\widetilde{v}_n,\frac{1}{n}\widetilde{w}_n)]\) is a solution of \(\big (GP_{(M,\textbf{0})}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\big )\). Therefore, by the maximum principle (Theorem 2.6),
Now, since \(\widetilde{v}_n\ll M\) in \(\Omega _1\) (recall Remark 2.8(iii)), we have that \(\widetilde{v}_n\le M\) and, consequently, also \(v_n\le M\). Hence, since \(M\le \widetilde{M}<\varGamma ^+ \),
but we also have
(B) For \(\varGamma ^-<0=\varGamma ^+\): let \(\varGamma ^-<m<\widetilde{m}< 0\), and \(v\in L^\infty (\Omega _1,\nu )\) be such that
As in the previous case, by the results in Subsect. 2.2.4, we know that, for \(n\in {\mathbb {N}}\), there exist \(u_n\in L^p(\Omega ,\nu )\), \(v_n\in L^{p'}(\Omega _1,\nu )\) and \(w_n\in L^{p'}(\Omega _2,\nu )\), such that \([u_n,(v_n,\frac{1}{n} w_n)]\) is a solution of \(\left( GP_{(v,\textbf{0})}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \). Then, since for the null function \(\textbf{0}\) in \(\Omega \), \([\textbf{0},\textbf{0}]\) is a solution of \(\left( GP_\textbf{0}^{\frac{1}{n}\textbf{a}_p,\gamma ,\beta }\right) \) and \(v<0\), the maximum principle yields
Therefore, in all the cases, \(\{u_n\}_n\) is \(L^\infty (\Omega ,\nu )\)-bounded from above. With a similar reasoning we obtain that, in any of these cases, \(\{u_n\}_n\) is also \(L^\infty (\Omega ,\nu )\)-bounded from below. Then, since
we obtain that
as desired. \(\square \)
The following theorem gives the existence and uniqueness of solutions of Problem \(\left( {\text {DP}}_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \). Recall that \(\varGamma ^-<\varGamma ^+\) and \(\mathfrak {B}^-<\mathfrak {B}^+\).
Theorem 3.10
Let \(T>0\). Let \(v_0\in L^{1}(\Omega _1,\nu )\) and \(f\in L^1(0,T;L^{1}(\Omega _1,\nu ))\). Assume
and
and
Then, there exists a unique mild-solution \(v\in C([0,T];L^1(\Omega _1,\nu ))\) of \(\displaystyle \left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \).
Let v and \({\widetilde{v}}\) be the mild solutions of the problem with respective data \(v_0,\ {\widetilde{v}}_0\in L^{1}(\Omega _1,\nu )\) and \(f,\ {\widetilde{f}}\in L^1(0,T;L^{1}(\Omega _1,\nu ))\). Then,
Under the additional assumptions
the mild solution v belongs to \(W^{1,1}(0,T;L^{1}(\Omega _1,\nu ))\) and satisfies the equation
that is, v is a strong solution.
The proof of this result differs, strongly at some points, from the proof of Theorem 3.4.
Proof
We start by proving the existence of mild solutions. For \(n\in {\mathbb {N}}\), consider the partition
where \(t_i^n:=iT/n\), \(i=1,\ldots ,n\). Given \(\epsilon >0\), since \(\mathfrak {B}^-<\mathfrak {B}^+\), there exist \(n\in \mathbb {N}\), \(v_0^n \in \overline{D(B^{m,\gamma ,\beta }_{\textbf{a}_p})}^{L^{p'}(\Omega _1,\nu )}\) (i.e., \(v_0^n\in L^{p'}(\Omega _1,\nu )\) satisfying \(\varGamma ^-\le v_0^n\le \varGamma ^+\)) and \(f_i^n \in L^{p'}(\Omega _1,\nu )\), \(i=1,\ldots n\), such that \(T/n\le \epsilon \),
and
Then, setting
we have that
Using the results in Subsect. 2.2.4, we see that, for n large enough, we may recursively find a solution \([u_i^n,(v_i^n,\frac{T}{n} w_i^n)]\) of \(\displaystyle \left( GP^{\frac{T}{n}\textbf{a}_p,\gamma ,\frac{T}{n}\beta }_{\left( \frac{T}{n} f_i^n+v_{i-1}^n,\textbf{0}\right) }\right) \), \(i=1,\ldots ,n\), so that
or, equivalently,
with \(v_i^n(x)\in \gamma (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _1\) and \(w_i^n(x)\in \beta (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _2\), \(i=1,\ldots ,n\). That is, we may find the unique solution \(v_i^n\) of the time discretization scheme associated with \(\left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \).
To apply these results, we must ensure that
holds for each step, but this holds true thanks to the choice of the \(f_i^n\), \(i=1,\ldots ,n\).
Therefore,
is an \(\epsilon \)-approximate solution of Problem \(\left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \). Consequently, by nonlinear semigroup theory ((see [11, 10, Theorem 4.1], or [8, Theorem A.27])) and on account of Theorem 3.7 and Theorem 3.9, we have that \(\left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \) has a unique mild solution \(v\in C([0,T];L^1(\Omega _1,\nu ))\) with
Uniqueness and the maximum principle for mild solutions is guaranteed by the T-accretivity of the operator.
We now prove, step by step, that these mild solutions are strong solutions of Problem \(\left( DP_{f,v_0}^{ \textbf{a}_p,\gamma ,\beta }\right) \) under the set of assumptions given in (3.21).
Let us define
and
Step 1. Suppose first that \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \).
In the construction of the mild solution, we now take \(v_0^n=v_0\) (since \(v_0\in L^{p'}(\Omega _1,\nu )\)) and the functions \(f_i^n \in L^{p'}(\Omega _1,\nu )\), \(i=1,\ldots n\), additionally satisfying
and
Multiplying both equations in (3.24) by \(u_i^n\), integrating with respect to \(\nu \) over \(\Omega _1\) and \(\Omega _2\), respectively, and adding them, we obtain
Then, since \(w_i^n(x)\in \beta (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _2\) the second term on the left-hand side is nonnegative and integrating by parts the third term, we get
Now, since \(v_i^n(x)\in \gamma (u_i^n(x))\) for \(\nu \)-a.e. \(x\in \Omega _1\),
Consequently,
Therefore, from (3.26), it follows that
\(i=1,\ldots ,n\). Then, integrating this equation over \(]t_{i-1},t_i]\) and adding for \(1\le i \le n\), we get
which, recalling the definitions of \(f_n\), \(u_n\), \(v_n\) and \(w_n\), can be rewritten as:
This, together with (2.5) and the fact that \(j^*_\gamma \) is nonnegative, yields
Therefore, for any \(\delta >0\), by (3.21) and Young’s inequality, there exists \(C(\delta )>0\) such that, in particular,
Observe also that, for any \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,n\}\), and for \(t\in ]t_{i-1}^n,t_i^n]\),
Indeed, multiplying the first equation in (3.23) by \(\frac{1}{r}T_r^+(u_i^n)\) and integrating with respect to \(\nu \) over \(\Omega _1\), then multiplying the second by \(\frac{T}{n}\frac{1}{r}T_r^+(u_i^n)\) and integrating with respect to \(\nu \) over \(\Omega _2\), adding both equations, neglecting the nonnegative term involving \(\textbf{a}_p\) (recall Remark 2.5) and letting \(r\downarrow 0\), we get that
i.e.,
Therefore,
which is equivalent to (3.29).
Now, by (3.25), if \(\varGamma ^+=+\infty \), there exists \(M>0\) such that
Consequently, Lemma A.7 applied for \(A=\Omega _1\), \(B=\emptyset \) and \(\alpha =\gamma \), yields
for every \(n\in {\mathbb {N}}\), every \(0\le t \le T\) and some constant \( K_2>0\).
Suppose now that \(\varGamma ^+<+\infty \). Then, by (3.29) we have that, for any \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,n\}\), and for \(t\in ]t_{i-1}^n,t_i^n]\) if \(i\ge 2\), or \(t\in [t_{0}^n,t_1^n]\) if \(i=1\),
thus, by the assumptions in (3.21) and by (3.22), there exists \(M\in {\mathbb {R}}\) such that
for n sufficiently large and, by (3.25), such that
for n sufficiently large. Therefore, we may apply Lemma A.8 for \(A=\Omega _1\), \(B=\emptyset \) and \(\alpha =\gamma \) to conclude that there exists a constant \(K_2'>0\) such that
for n sufficiently large.
Similarly, we may find \( K_3>0\) such that
for n sufficiently large.
Consequently, by the generalised Poincaré-type inequality together with (3.28) for \(\delta \) small enough, we get
for some constant \(K_4>0\), that is, \(\{u_n\}_n\) is bounded in \(L^p(0,T;L^p(\Omega ,\nu ))\). Therefore, there exists a subsequence, which we continue to denote by \(\{u_{n}\}_{n}\), and \(u\in L^p(0,T;L^p(\Omega ,\nu ))\) such that
Note that, since \( \displaystyle \Big \{\int _0^T\int _\Omega \int _\Omega |u_n(t)(y)-u_n(t)(x)|^p dm_x(y){\text {d}}\nu (x){\text {d}}t\Big \}_n \) is bounded, then, by (2.4), we have that \(\{[(t,x,y)\mapsto \textbf{a}_p(x,y,u_n(t)(y)-u_n(t)(x))]\}_n\) is bounded in \(L^{p'}(0,T; L^{p'}(\Omega \times \Omega ,\nu \otimes m_x))\) so we may take a further subsequence, which we continue to denote in the same way, such that
Now, let \(\Psi \in W^{1,1}_0(0,T;L^p(\Omega ,\nu ))\), \(\hbox {supp}(\Psi )\subset \subset [0,T]\), then
for \(x\in \Omega _1\). Therefore, multiplying both equations in (3.24) by \(\Psi \), integrating the first one over \(\Omega _1\) and the second one over \(\Omega _2\) with respect to \(\nu \), adding them, and taking limits as \(n\rightarrow +\infty \) we get that
Therefore, taking \(\Psi (t)(x)=\psi (t)\xi (x)\), where \(\psi \in C_c^\infty (0,T)\) and \(\xi \in L^p(\Omega ,\nu )\), we obtain that
for \(\nu \)-a.e. \(x\in \Omega _1\).
It follows that
Therefore, since \(v\in C([0,T];L^1(\Omega _1,\nu ))\), \(\Phi \in L^{p'}(0,T; L^{p'}(\Omega \times \Omega ,\nu \otimes m_x))\) and \(f\in L^{p'}(0,T;L^{p'} (\Omega _1,\nu ))\), we get that \(v'\in L^{p'}(0,T;L^{p'}(\Omega _1,\nu ))\) and \(v\in W^{1,1}(0,T;L^{1}(\Omega _1,\nu ))\).
Then, by Remark 3.8, we conclude that the mild solution v is, in fact, a strong solution (see [11] or [8, Corollary A.34]). Hence,
Let us see, for further use, that \(\displaystyle \int _{\Omega _1}j_{\gamma }^*(v(t)){\text {d}}\nu \in W^{1,1}(0,T)\). By (3.27) and Fatou’s lemma, we have
Moreover, by (3.30),
where F is given by
Let \(\psi \in C_c^\infty (0,T)\), \(\psi \ge 0\), \(\tau >0\) and
Then, for \(\tau \) small enough, \(\eta _\tau \in W^{1,1}_0(0,T;L^p(\Omega _1,\nu ))\) and we may use it as a test function in (3.31) to obtain
Now, since
which, letting \(\tau \rightarrow 0^+\) yields
Taking
yields the opposite inequalities so that, in fact,
i.e.,
thus, in particular,
Step 2. Suppose now that, either \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+<+\infty \), or \(\mathcal {R}_{\gamma ,\beta }^->-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \). Recall that we are assuming the hypotheses in (3.21) and that \(v_0^n=v_0\) for every \(n\in {\mathbb {N}}\). Suppose first that \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+<+\infty \). Then, for \(k\in \mathbb {N}\), let \(\beta ^k:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be the following maximal monotone graph:
We have that \(\beta ^k\rightarrow \beta \) in the sense of maximal monotone graphs. Indeed, given \(\lambda >0\) and \(s\in {\mathbb {R}}\), there exists \(r\in {\mathbb {R}}\) such that \(s\in r+\lambda \beta (r)\) thus, for \(k>r\), \(s\in r+\lambda \beta (r)= r+\lambda \beta ^k(r)\), i.e., \(r=(I+\lambda \beta )^{-1}(s)=(I+\lambda \beta ^k)^{-1}(s)\).
By Step 1, we know that, since \(\mathcal {R}_{\gamma ,\beta ^k}^-=-\infty \) and \(\mathcal {R}_{\gamma ,\beta ^k}^+=+\infty \), there exists a strong solution \(v_k\in W^{1,1}(0,T;L^{1}(\Omega _1,\nu ))\) of Problem \(\left( DP_{f-\frac{1}{k},v_0}^{ \textbf{a}_p,\gamma ,\beta ^k}\right) \), therefore, there exist \(u_k\in L^p(0,T;L^p(\Omega ,\nu ))\) and \(w_k\in L^{p'}(0,T;L^{p'}(\Omega _2,\nu ))\) such that
with \(v_k\in \gamma (u_k)\) \(\nu \)-a.e. in \(\Omega _1\) and \(w_k\in \beta ^k(u_k)\) \(\nu \)-a.e. in \(\Omega _2\). Let us see that
and
Going back to the construction of the mild solution, in this case of \(\left( DP_{f-\frac{1}{k},v_0}^{ \textbf{a}_p,\gamma ,\beta ^k}\right) \), for each step \(n\in {\mathbb {N}}\) and for each \(i\in \{1,\ldots , n\}\), we have that there exists \(u_{k,i}^n\in L^p(\Omega ,\nu )\), \(v_{k,i}^n\in L^{p'}(\Omega _1,\nu )\) and \(w_{k,i}^n\in L^{p'}(\Omega _2,\nu )\) such that
with \(v_{k,i}^n\in \gamma (u_{k,i}^n)\) \(\nu \)-a.e. in \(\Omega _1\) and \(w_{k,i}^n\in \beta ^k(u_{k,i}^n)\) \(\nu \)-a.e. in \(\Omega _2\). Let
for \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,n\}\) (observe that \(\beta ^k(r)\) is single-valued for \(r>k\) and coincides with \(\beta ^{k+1}(r)=\beta (r)\) for \(r<k\)). It is clear that \(z_{k,i}^n\in \beta ^k(u_{k+1,i}^n)\) and, since \(\beta ^k\ge \beta ^{k+1}\), \(z_{k,i}^n\ge w_{k+1,i}^n\). Then,
for \(n\in {\mathbb {N}}\). Hence, by the maximum principle (Theorem 2.6),
Proceeding in the same way, we get that
for each \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots , n\}\). From here we get (3.34) and (3.35).
Since \(\gamma ^{-1}(r)= \partial j_{\gamma }^*(r)\) and \(u_k(t)\in \gamma ^{-1}(v_k(t))\) \(\nu \)-a.e. in \(\Omega _1\), we have
Integrating this equation over [0, T], dividing by \(\tau \), letting \(\tau \rightarrow 0^+\) and recalling that, by (3.32), \(\displaystyle \int _{\Omega _1}j_\gamma ^*(v_k){\text {d}} \nu \in W^{1,1}(0,T)\), we get
Therefore, multiplying (3.33) by \(u_k\) and integrating with respect to \(\nu \), we get
Now, working as in the previous step, since \(\Gamma ^{+}<\infty \), we get that \( \displaystyle \left\{ \Vert u_k \Vert _{L^{p}(0,T;L^p(\Omega ,\nu ))}^{p}\right\} _k\) is bounded. Then, by the monotone convergence theorem, we get that there exists \(u\in L^p(0,T;L^p(\Omega ,\nu ))\) such that \(u_k{\mathop {\longrightarrow }\limits ^{k}}u\) in \(L^p(0,T;L^p(\Omega ,\nu ))\). From this we get, by [15, Lemma G], that \(v(t)(x)\in \gamma (u(t)(x))\) for a.e. \(t\in [0,T]\) and \(\nu \)-a.e. \(x\in \Omega _1\).
Therefore, from (3.33) and Lemma 2.1 (note that, by the monotonicity of \(\{u_k\}\), \(|u_k|\le \max \{|u_1|,|u|\}\in L^p(\Omega ,\nu )\)), we get that \((v_k)_t\) converges strongly in \(L^{p'}(0,T;L^{p'}(\Omega _1,\nu ))\) and \(w_k\) converges strongly in \(W^{1,1}(0,T;L^{1}(\Omega _1,\nu ))\). In particular, \(v\in W^{1,1} (0,T;L^1(\Omega _1,\nu ))\), \(w(t)(x)\in \beta (u(t)(x))\) for a.e. \(t\in [0,T]\) and \(\nu \)-a.e. \(x\in \Omega _2\), and
The case \(\mathcal {R}_{\gamma ,\beta }^->-\infty \) and \(\mathcal {R}_{\gamma ,\beta }^+=+\infty \) follows similarly by taking
instead of \(\beta ^k\), \(k\in {\mathbb {N}}\).
Step 3. Finally, assume that both \(\mathcal {R}_{\gamma ,\beta }^-\) and \(\mathcal {R}_{\gamma ,\beta }^+\) are finite. We define, for \(k\in \mathbb {N}\),
By the previous step, we have that for k large enough such that \(f+\frac{1}{k}\) satisfies
there exists a strong solution \(v_k\in W^{1,1}(0,T;L^{1}(\Omega _1,\nu ))\) of Problem \(\left( DP_{f+\frac{1}{k},v_0}^{ \textbf{a}_p,\gamma ,{\widetilde{\beta }}^k}\right) \), i.e., there exist \(u_k\in L^p(0,T;L^p(\Omega ,\nu ))\) and \(w_k\in L^{p'}(0,T;L^{p'}(\Omega _2,\nu ))\) such that
with \(v_k\in \gamma (u_k)\) \(\nu \)-a.e. in \(\Omega _1\) and \(w_k\in {\tilde{\beta }}^k(u_k)\) \(\nu \)-a.e. in \(\Omega _2\).
Going back to the construction of the mild solution, in this case of \(\left( DP_{f+\frac{1}{k},v_0}^{ \textbf{a}_p,\gamma ,{\widetilde{\beta }}^k}\right) \), for each step \(n\in {\mathbb {N}}\) and for each \(i\in \{1,\ldots , n\}\), we have that there exists \(u_{k,i}^n\in L^p(\Omega ,\nu )\), \(v_{k,i}^n\in L^{p'}(\Omega _1,\nu )\) and \(w_{k,i}^n\in L^{p'}(\Omega _2,\nu )\) such that
where \(v_{k,i}^n\in \gamma (u_{k,i}^n)\) \(\nu \)-a.e. in \(\Omega _1\) and \(w_{k,i}^n\in {\widetilde{\beta }}^k(u_{k,i}^n)\) \(\nu \)-a.e. in \(\Omega _2\). Let
for \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,n\}\) (observe that \({\widetilde{\beta }}^k(r)\) is single-valued for \(r<-k\) and coincides with \({\widetilde{\beta }}^{k+1}(r)=\beta (r)\) for \(r>-k\)). It is clear that \(z_{k,i}^n\in {\widetilde{\beta }}^k(u_{k+1,i}^n)\) and, since \({\widetilde{\beta }}^k\le {\widetilde{\beta }}^{k+1}\), we have that \(z_{k,i}^n\le w_{k+1,i}^n\), \(i\in \{1,\ldots ,n\}\). Then,
for \(n\in {\mathbb {N}}\). Hence, by the maximum principle (Theorem 2.6),
Proceeding in the same way we get that, for \(n\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,n\}\),
Therefore,
and
We can now conclude, as in the previous step, that
for some constant \( K_5>0\). Moreover, by the monotonicity of \(\{u_k\}\), we get that \(\displaystyle \Big \{\int _0^T\Vert u_k^+(t)\Vert _{L^{p}(\Omega _1,\nu )}{\text {d}}t\Big \}_k\) is bounded. From this point, we can finish the proof as in the previous step. \(\square \)
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Acknowledgements
The authors appreciate the suggestions and comments made by the anonymous referee, which have allowed for a better presentation of this work. The authors are grateful to J. M. Mazón for stimulating discussions on this paper. The authors have been partially supported by the Spanish MICIU and FEDER, project PGC2018-094775-B-100, and by the “Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital”, project AICO/2021/223. The first author was also supported by the Spanish MICIU under grant BES-2016-079019 (which is supported by the European FSE); by the Spanish Ministerio de Universidades and NextGenerationUE, programme “recualificación del sistema universitario español” (Margarita Salas) under grant UP2021-044; and the Conselleria d’Innovació, Universitats, Ciència i Societat Digital, programme “Subvenciones para la contratación de personal investigador en fase postdoctoral” (APOSTD 2022), under grant CIAPOS/2021/28.
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Spanish MICIU and FEDER, project PGC2018-094775-B-100. Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital, project AICO/2021/223. Spanish MICIU and European FSE, grant BES-2016-079019. Ministerio de Universidades and NextGenerationUE, programme “recualificación del sistema universitario español” (Margarita Salas), Ref. UP2021-044. Conselleria d’Innovació, Universitats, Ciència i Societat Digital, programme “Subvenciones para la contratación de personal investigador en fase postdoctoral” (APOSTD 2022), Ref. CIAPOS/2021/28.
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A Poincaré-type inequalities
A Poincaré-type inequalities
In order to prove the results on existence of solutions of our problems, we have assumed that appropriate Poincaré-type inequalities hold. In [47, Corollary 31], it is proved that a Poincaré-type inequality holds on metric random walk spaces (with an invariant measure) with positive coarse Ricci curvature. Under some conditions relating the random walk and the invariant measure some Poincaré-type inequalities are given in [42, Theorem 4.5] (see also [8, 43]). Here, we generalise some of these results.
Definition A.1
Let [X, d, m] be a metric random walk space with reversible measure \(\nu \) and let \(A, B\subset X\) be disjoint measurable sets such that \(\nu (A)>0\). Let \(Q:=((A\cup B)\times (A\cup B)){\setminus } (B\times B)\). We say that [X, d, m] satisfies a generalised (q, p)-Poincaré -type inequality (\(p, q\in [1,+ \infty [\)) on (A, B) (with respect to \(\nu \)), if, given \(0<l\le \nu (A\cup B)\), there exists a constant \(\Lambda >0\) such that, for any \(u \in L^q(A\cup B,\nu )\) and any measurable set \(Z\subset A\cup B\) with \(\nu (Z)\ge l\),
In Subsect. 2.2 (Assumption 5), we have used that the metric random walk space satisfies a generalised (p, p)-Poincaré-type inequality on \((\Omega _1\cup \Omega _2,\emptyset )\). This assumption holds true in many important examples, as the next results show.
Lemma A.2
Let [X, d, m] be a metric random walk space with reversible measure \(\nu \) with respect to m. Let \(A, B\subset X\) be disjoint measurable sets such that \(B\subset \partial _m A\), \(\nu (A)>0\) and A is m-connected. Suppose that \(\nu (A\cup B)<+\infty \) and that
Let \(q\ge 1\). Let \(\{u_n\}_n\subset L^q(A\cup B,\nu )\) be a bounded sequence in \(L^1(A\cup B,\nu )\) satisfying
where, as before, \(Q=((A\cup B)\times (A\cup B)){\setminus }(B\times B)\). Then, there exists \(\lambda \in \mathbb {R}\) such that
and
Proof
If \(B=\emptyset \) (or \(\nu \)-null), one can skip some steps in the proof. Let
and
Let
From (A.1), it follows that
and
Passing to a subsequence if necessary, we can assume that
and
On the other hand, by (A.1), we also have that
Therefore, we can suppose that, up to a subsequence,
Let \(N_1\subset A\) be a \(\nu \)-null set satisfying that,
and \(N_2\subset A\cup B\) be a \(\nu \)-null set satisfying that,
Now, since A is m-connected and \(B\subset \partial _m A\),
is \(\nu \)-null. Indeed, by the definition of D, \(L_m(A\cap D, A)=0\) thus, in particular, \(L_m(A\cap D,A{\setminus } D)=0\) which, since A is m-connected, implies that \(\nu (A\cap D)=0\) or \(\nu (A\cap D)=\nu (A)\). However, if \(\nu (A\cap D)=\nu (A)\), then for any E, \(F\subset A\), we have \(L_m(E,F)\le L_m(D\cap A,A)=0\) which is a contradiction, thus \(\nu (D\cap A)=0\). Now, since \(B\subset \partial _m A\), \(m_x(A)>0\) for every \(x\in B\), thus \(\nu (B\cap D)=0\).
Set \(N:=\mathcal {N}_\perp \cup N_f\cup N_g\cup N_1\cup N_2\cup D\) (note that \(\nu (N)=0\)). Fix \(x_0\in A\setminus N\). Up to a subsequence, \(u_n(x_0)\rightarrow \lambda \) for some \(\lambda \in [-\infty ,+\infty ]\), but then, by (A.4), we also have that \(u_n(y)\rightarrow \lambda \) for every \(y\in (A\cup B){\setminus } C_{x_0}\). However, since \(x_0\not \in \mathcal {N}_\perp \) and \(m_{x_0}(C_{x_0})=0\), we must have that \(\nu (A\setminus C_{x_0})>0\); thus, if
then \(\nu (S\cap A)\ge \nu (A\setminus C_{x_0})>0\). Note that, if \(x\in (A\cap S){\setminus } N\) then, by (A.4) again, \((A\cup B){\setminus } C_x\subset S\) thus \(m_x((A\cup B){\setminus } S)\le m_x(C_x)=0\); therefore,
In particular, \(L_m(A\cap S, A\setminus S)=0\), but, since A is m-connected and \(\nu (A\cap S)>0\), we must have \(\nu (A{\setminus } S)=0\), i.e. \(\nu (A)=\nu (A\cap S)\).
Finally, suppose that \(\nu (B\setminus S)>0\). Let \(x\in B{\setminus } (S\cup N)\). By (A.4), \(A{\setminus } C_x'\subset A{\setminus } S\), i.e., \(S\cap A\subset C_x'\), thus \(m_x(S\cap A)=0\). Therefore, since \(x\not \in \mathcal {N}_\perp \), we must have \(\nu (A{\setminus } S)>0\) which is a contradiction with what we have already obtained. Consequently, we have obtained that \(u_n\) converges \(\nu \)-a.e. in \(A\cup B\) to \(\lambda \):
Since \(\{\Vert u_n\Vert _{L^1 (A\cup B,\nu )}\}_n\) is bounded, by Fatou’s lemma, we must have that \(\lambda \in {\mathbb {R}}\). On the other hand, by (A.2),
for every \(x\in \Omega \setminus N_f\). In other words, \(\Vert u_n(\cdot )-u_n(x)\Vert _{L^q (A, m_x)}\rightarrow 0\), thus
Similarly, by (A.3),
\(\square \)
Theorem A.3
Let \(p\ge 1\). Let [X, d, m] be a metric random walk space with reversible measure \(\nu \). Let \(A,B\subset X\) be disjoint measurable sets such that \(B\subset \partial _m A\), \(\nu (A)>0\) and A is m-connected. Suppose that \(\nu (A\cup B)<+\infty \) and that
Assume further that, given a \(\nu \)-null set \(N\subset A\), there exist \(x_1,x_2,\ldots , x_L\in A\setminus N\) and a constant \(C>0\) such that 
. Then, [X, d, m] satisfies a generalised (p, p)-Poincaré-type inequality on (A, B).
Proof
Let \(p\ge 1\) and \(0<h\le \nu (A\cup B)\). We want to prove that there exists a constant \(\Lambda >0\) such that
for every \(u \in L^p(A\cup B,\nu )\) and every measurable set \(Z\subset A\cup B\) with \(\nu (Z)\ge l\). Suppose that this inequality is not satisfied for any \(\Lambda \). Then, there exists a sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset L^p(A\cup B,\nu )\), with \(\Vert u_n\Vert _{L^p (A\cup B,\nu )}=1\), and a sequence of measurable sets \(Z_n\subset A\cup B\) with \(\nu (Z_n)\ge l\), \(n\in {\mathbb {N}}\), satisfying
and
Therefore, by Lemma A.2, there exist \(\lambda \in \mathbb {R}\) and a \(\nu \)-null set \(N\subset A\) such that
Now, by hypothesis, there exist \(x_1,x_2,\ldots , x_L\in A{\setminus } N\) and \(C>0\) such that 
. Therefore,
Moreover, since 
is bounded in \(L^{p'}(A\cup B, \nu )\), there exists \(\phi \in L^{p'}(A\cup B,\nu )\) such that, up to a subsequence, 
weakly in \(L^{p'}(A\cup B,\nu )\) (weakly-\(*\) in \(L^\infty (A\cup B,\nu )\) in the case \(p=1\)). In addition, \(\phi \ge 0\) \(\nu \)-a.e. in \(A\cup B\) and
Then, since \(u_n{\mathop {\longrightarrow }\limits ^{n}}\lambda \) in \(L^p(A\cup B,\nu )\) and 
weakly in \(L^{p'}(A\cup B,\nu )\) (weakly-\(*\) in \(L^\infty (A\cup B,\nu )\) in the case \(p=1\)),
thus \(\lambda =0\). This is a contradiction with \(||u_n||_{L^p (A\cup B,\nu )}=1\), \(n\in {\mathbb {N}}\), since \(u_n{\mathop {\longrightarrow }\limits ^{n}} \lambda \) in \(L^p(A\cup B,\nu )\), so the theorem is proved. \(\square \)
Remark A.4
If \(\Omega :=\Omega _1\cup \Omega _2\) is m-connected we can apply the theorem with \(A:=\Omega \) and \(B=\emptyset \) to obtain the generalised Poincaré-type inequality used in Subsect. 2.2 (Assumption 5).
We can take \(A=X\), \(B=\emptyset \) and \(Z=X\) in the theorem to obtain [42, Theorem 4.5].
Remark A.5
The assumption that given a \(\nu \)-null set \(N\subset A\), there exist \(x_1,x_2,\ldots , x_L\in A{\setminus } N\) and \(C>0\) such that 
is not as strong as it seems. Indeed, this is trivially satisfied by connected locally finite weighted discrete graphs and is also satisfied by \([{\mathbb {R}}^N,d,m^J]\) (recall Examples 1.1 and 1.2) if, for a domain \(A\subset {\mathbb {R}}^N\), we take \(B\subset \partial _{m^J} A\) such that \(dist(B,{\mathbb {R}}^N{\setminus } A_{m^J})>0\). Moreover, in the following example we see that if we remove this hypothesis then the statement is not true in general.
Consider the metric random walk space \([{\mathbb {R}},d,m^J]\) where d is the Euclidean distance and 
(recall Example 1.2). Let \(A:=[-1,1]\) and \(B:=\partial _{m^J}A =[-2,2]{\setminus } A\). Then, if \(N=\{-1,1\}\) we may not find points satisfying the aforementioned assumption. In fact, the statement of the theorem does not hold for any \(p>1\) as can be seen by taking 
and \(Z:=A\cup B\). Indeed, first note that \(\Vert u_n\Vert _{ L^p([-2,2],\nu )}=1\) and \(\int _{[-2,2]}u_n{\text {d}}\nu =0\) for every \(n\in {\mathbb {N}}\). Now, \(\hbox {supp}(m^J_x)=[x-1,x+1]\) for \(x\in [-1,1]\) and, therefore,
for \(x\in [-1,1]\). Consequently,
Finally, by the reversibility of \(\mathcal {L}^1\) with respect to \(m^J\),
thus
However, in this example, as we mentioned before, we can take \(B\subset \partial _m A\) such that \(\text{ dist }(B,{\mathbb {R}}{\setminus } [-2,2])>0\) to avoid this problem and to ensure that the hypotheses of the theorem are satisfied so that (A, B) satisfies a generalised (p, p)-Poincaré-type inequality.
In the following example, the metric random walk space [X, d, m] that is defined, together with the invariant measure \(\nu \), satisfies that \(m_x\perp \nu \) for every \(x\in X\), and a Poincaré-type inequality does not hold.
Example A.6
Let \(p>1\). Let \(S^1=\{e^{2\pi i \alpha } \,: \, \alpha \in [0,1)\}\) and let \(T_{\theta }:S^1\longrightarrow S^1\) denote the irrational rotation map \(T_{\theta }(x)=xe^{2\pi i \theta }\) where \(\theta \) is an irrational number. On \(S^1\) consider the Borel \(\sigma \)-algebra \(\mathcal {B}\) and the 1-dimensional Hausdorff measure 
. It is well known that \(T_{\theta }\) is a uniquely ergodic measure-preserving transformation on \((S^1,\mathcal {B},\nu )\).
Now, denote \(X:=S^1\) and let \(m_x:=\frac{1}{2} \delta _{T_{-\theta } (x)}+\frac{1}{2} \delta _{T_{\theta }(x)}\), \(x\in X\). Then, \(\nu \) is reversible with respect to the metric random walk space [X, d, m], where d is the metric given by the arclength. Indeed, let \(f\in L^1(X\times X, \nu \otimes \nu )\), then
Let us see that this space is m-connected. First note that, for \(x\in X\),
and, by induction, it is easy to see that
Here, \(m_x^{*n}\), \(n\in {\mathbb {N}}\), is defined inductively as follows (see [41]):
Now, let \(A\subset X\) such that \(\nu (A)>0\). By the pointwise ergodic theorem,
for \(\nu \)-a.e. \(x\in X\). Consequently, for \(\nu \)-a.e. \(x\in X\), there exists \(n\in {\mathbb {N}}\) such that
thus \(\nu \left( \left\{ x\in X\,: \, m_x^{*n}(A)=0 \ \hbox { for every } n\in {\mathbb {N}}\right\} \right) =0\). Then, according to [41, Definition 2.8] (see also [41, Proposition 2.11]), [X, d, m] with the invariant measure \(\nu \) for m is m-connected.
Let us see that \([X,d,m,\nu ]\) does not satisfy a (p, p)-Poincaré-type inequality. For \(n\in {\mathbb {N}}\) let
where \(\delta (n)>0\) is chosen so that
(note that \(e^{2\pi i (k_1\theta -\delta (n))}\ne e^{2\pi i (k_2\theta -\delta (n))}\) for every \(k_1\ne k_2\) since \(T_\theta \) is ergodic). Consider the following sequence of functions:
Then,
and
Fix \(n\in {\mathbb {N}}\), let us see what happens with
If \(1\le k\le n-2\) or \(n+1\le k\le 2n-2\) and \(x\in I_k^n\), then
since \(T_{-\theta }(x)\in I_{k-1}^n\) and \(T_{\theta }(x)\in I_{k+1}^n\). Now, if \(x\in I_0^n\) then \(T_{-\theta }(x)\in I_{-1}^n\) thus
and the same holds if \(x\in I_{2n-1}^n\) (then \(T_{\theta }(x)\in I_{2n}^n\)). For \(x\in I_{n-1}\) we have \(T_{\theta }(x)\in I_{n}^n\) thus
and the same result is obtained for \(x\in I_{n+1}^n\). Similarly, if \(x\in I_{-1}^n\) or \(x\in I_{2n}^n\),
Finally, if \(x\notin \cup _{k=-1}^{2n}I_k^n\), then \(T_{-\theta }(x), T_{\theta }(x)\not \in \cup _{k=0}^{2n-1}I_k^n\) thus
Consequently,
Therefore, there is no \(\Lambda >0\) such that
since this would imply
The proofs of the following lemmas are similar to the proof of [4, Lemma 4.2].
Lemma A.7
Let \(p\ge 1\). Let [X, d, m] be a metric random walk space with reversible measure \(\nu \) with respect to m. Let \(A, B\subset X\) be disjoint measurable sets and assume that \(A\cup B\) is non-\(\nu \)-null and m-connected. Suppose that [X, d, m] satisfies a generalised (p, p)-Poincaré-type inequality on \((A\cup B,\emptyset )\). Let \(\alpha \) and \(\tau \) be maximal monotone graphs in \({\mathbb {R}}^2\) such that \(0\in \alpha (0)\) and \(0\in \tau (0)\). Let \(\{u_n\}_{n\in {\mathbb {N}}}\subset L^p(A\cup B,\nu )\), \(\{z_n\}_{n\in {\mathbb {N}}}\subset L^1(A,\nu )\) and \(\{\omega _n\}_{n\in {\mathbb {N}}}\subset L^1(B,\nu )\) be such that, for every \(n\in {\mathbb {N}}\), \(z_n\in \alpha (u_n)\) \(\nu \)-a.e. in A and \(\omega _n\in \tau (u_n)\) \(\nu \)-a.e. in B.
-
(i)
Suppose that \(\mathcal {R}_{\alpha ,\tau }^+=+\infty \) and that there exists \(M>0\) such that
$$\begin{aligned} \int _{A}z_n^+{\text {d}}\nu +\int _{B}\omega _n^+{\text {d}}\nu <M \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$Then, there exists a constant \(K=K(A,B,M,\alpha , \tau )\) such that
$$\begin{aligned}{} & {} \left\| u^+_n \right\| _{L^p(A\cup B,\nu )} \le K\left( \left( \int _{(A\cup B)\times (A\cup B)} |u^+_n(y)-u^+_n(x)|^p dm_x(y) {\text {d}}\nu (x) \right) ^{\frac{1}{p}}+1\right) \\ {}{} & {} \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$ -
(ii)
Suppose that \(\mathcal {R}_{\gamma ,\beta }^-=-\infty \) and that there exists \(M>0\) such that
$$\begin{aligned} \int _{A}z_n^-{\text {d}}\nu +\int _{B}\omega _n^-{\text {d}}\nu <M \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$Then, there exists a constant \(\widetilde{K}=\widetilde{K}(A,B,M, \alpha ,\tau )\), such that
$$\begin{aligned} \left\| u^-_n \right\| _{L^p(A\cup B,\nu )} \le{} & {} \widetilde{K}\left( \left( \int _{(A\cup B)\times (A\cup B)} |u^-_n(y)-u^-_n(x)|^p dm_x(y) {\text {d}}\nu (x) \right) ^{\frac{1}{p}}+1\right) \\{} & {} \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$
Lemma A.8
Let \(p\ge 1\). Let [X, d, m] be a metric random walk space with reversible measure \(\nu \) with respect to m. Let \(A,B\subset X\) be disjoint measurable sets and assume that \(A\cup B\) is non-\(\nu \)-null and m-connected. Suppose that [X, d, m] satisfies a generalised (p, p)-Poincaré-type inequality on \((A\cup B,\emptyset )\). Let \(\alpha \) and \(\tau \) be maximal monotone graphs in \({\mathbb {R}}^2\) such that \(0\in \alpha (0)\) and \(0\in \tau (0)\). Let \(\{u_n\}_{n\in {\mathbb {N}}}\subset L^p(A\cup B,\nu )\), \(\{z_n\}_{n\in {\mathbb {N}}}\subset L^1(A,\nu )\) and \(\{\omega _n\}_{n\in {\mathbb {N}}}\subset L^1(B,\nu )\) such that, for every \(n\in {\mathbb {N}}\), \(z_n\in \alpha (u_n)\) \(\nu \)-a.e. in A and \(\omega _n\in \tau (u_n)\) \(\nu \)-a.e. in B.
-
(i)
Suppose that \(\mathcal {R}_{\alpha ,\tau }^+<+\infty \) and that there exists \(M\in {\mathbb {R}}\) and \(h>0\) such that
$$\begin{aligned} \int _{A}z_n{\text {d}}\nu +\int _{B}\omega _n{\text {d}}\nu<M<\mathcal {R}_{\alpha ,\tau }^+ \quad \hbox {for every } n\in {\mathbb {N}}, \end{aligned}$$and
$$\begin{aligned}{} & {} \max \left\{ \int _{\{x\in A \,:\, z_n<-h\}}|z_n|{\text {d}}\nu ,\int _{x\in B \,:\, \omega _n(x)<-h\}}|\omega _n|{\text {d}}\nu \right\} <\frac{\mathcal {R}_{\alpha ,\tau }^+-M}{8} \\ {}{} & {} \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$Then, there exists a constant \(K=K(A, B,M,h,\alpha , \tau )\) such that
$$\begin{aligned}{} & {} \left\| u^+_n \right\| _{L^p(A\cup B,\nu )} \le K\left( \left( \int _{(A\cup B)\times (A\cup B)} |u^+_n(y)-u^+_n(x)|^p dm_x(y) {\text {d}}\nu (x) \right) ^{\frac{1}{p}}+1\right) \\ {}{} & {} \quad \hbox {for every } n\in {\mathbb {N}}\end{aligned}$$ -
(ii)
Suppose that \(\mathcal {R}_{\alpha ,\tau }^->-\infty \) and that there exists \(M\in {\mathbb {R}}\) and \(h>0\) such that
$$\begin{aligned} \int _{A}z_n{\text {d}}\nu +\int _{B}\omega _n{\text {d}}\nu>M>\mathcal {R}_{\alpha ,\tau }^- \quad \hbox {for every } n\in {\mathbb {N}}, \end{aligned}$$and
$$\begin{aligned} \max \left\{ \int _{\{x\in A \,:\, z_n>h\}} z_n {\text {d}}\nu ,\int _{x\in B \,:\, \omega _n(x)>h\}}\omega _n {\text {d}}\nu \right\} <\frac{M-\mathcal {R}_{\alpha ,\tau }^-}{8} \quad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$Then, there exists a constant \(\widetilde{K}=\widetilde{K}(A,B,M,h, \alpha , \tau )\) such that
$$\begin{aligned}{} & {} \left\| u^-_n \right\| _{L^p(A\cup B,\nu )} \le \widetilde{K}\left( \left( \int _{(A\cup B)\times (A\cup B)} |u^-_n(y)-u^-_n(x)|^p dm_x(y) {\text {d}}\nu (x) \right) ^{\frac{1}{p}}+1\right) \\ {}{} & {} \quad \qquad \qquad \qquad \hbox {for every } n\in {\mathbb {N}}. \end{aligned}$$
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Solera, M., Toledo, J. Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions. J. Evol. Equ. 23, 24 (2023). https://doi.org/10.1007/s00028-022-00854-y
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DOI: https://doi.org/10.1007/s00028-022-00854-y
Keywords
- Random walks
- Nonlocal operators
- Weighted graphs
- \(p-\)Laplacian
- Neumann boundary conditions
- Diffusion in porous media
- Stefan problem
- Hele–Shaw problem
- Obstacle problems
- Dynamical boundary conditions