Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

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 RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N$$\end{document} 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.


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 ν for the random walk m (see Subsection 1.1 for details). The nonlocal diffusion can hold either in W , in its nonlocal boundary ∂mW , or in both at the same time. We will assume that W ∪ ∂mW is m-connected and ν-finite. The formulations of the diffusion problems that we study are the following x ∈ W, 0 < t < T, −N ap 1 u(t, x) ∈ β u(t, x) , x ∈ ∂mW, 0 < t < T, v(0, x) = v 0 (x), x ∈ W, (1.1) and, for nonlinear dynamical boundary conditions, x) = f (t, x), x ∈ W, 0 < t < T, v(t, x) ∈ γ u(t, x) , x ∈ W, 0 < t < T, w t (t, x) + N ap 1 u(t, x) = g(t, x), x ∈ ∂mW, 0 < t < T, w(t, x) ∈ β u(t, x) , x ∈ ∂mW, 0 < t < T, v(0, x) = v 0 (x), x ∈ W, w(0, x) = w 0 (x), x ∈ ∂mW, (1.2) where γ and β are maximal monotone (multivalued) graphs in R × R, divmap is a nonlocal Leray-Lions type operator whose model is the nonlocal p-Laplacian type diffusion operator, and N ap 1 is a nonlocal Neumann boundary operator (see Subsection 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 ∂mW , but rather for any two disjoint subsets Ω 1 and Ω 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 γ and β (see for example [4] and [15] for local problems). In [8], and the references therein, one can find an interpretation of the nonlocal diffusion process involved in these kind of problems. On the nonlinearities (brought about by) γ and β we do not impose any further assumptions aside from the natural one (see Bénilan, Crandall and Sacks [15]): 0 ∈ γ(0) ∩ β(0), and (in order for diffusion to take place) where Γ − = inf Ran(γ), Γ + = sup Ran(γ), B − = inf Ran(β) and B + = sup Ran(β).
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 ∂mW , or in both at the same time); the nonlocal counterpart of Stefan like problems, that involve monotone graphs like the graph inverse of [0, λ] if r = 0, λ + r if r > 0, for λ > 0; diffusion problems in porous media, where monotone graphs like ps(r) = |r| s−1 r, s > 0, are involved; and Hele-Shaw type problems, which involve graphs like Moreover, if γ = 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 (β = {0} × R) and the Neumann boundary condition (β = R × {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] and [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 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 Subsection 1.1, in particular Examples 1.1 and 1.2, and Definition 1.4, for the necessary definitions and notations): wx,y|u(y) − u(x)| p−2 (u(y) − u(x)), x ∈ W, 0 < t < T, v(t, x) ∈ γ u(t, x) , x ∈ W, 0 < t < T, 1 dx y∈W m G wx,y|u(y) − u(x)| p−2 (u(y) − u(x)) ∈ β(u(t, x)), x ∈ ∂ m G W, 0 < t < T, u(x, 0) = u 0 (x), x ∈ W, for weighted discrete graphs, and x ∈ W, 0 < t < T, for the case of 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 Section 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 γ u(x)) − divmapu(x) ∋ ϕ(x), x ∈ W, N ap 1 u(x) + β u(x) ∋ ϕ(x), x ∈ ∂mW, (1.3) for general maximal monotone graphs γ and β. 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 Section 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 Ω in R N , γ with bounded domain [0, 1] and β(r) = 0 for all r, it is not possible to find a weak solution of −∆u + γ(u) ∋ ϕ in Ω, for data satisfying ϕ ≤ 0, ϕ ≤ 0 and ϕ ≡ 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 is an extension to R of the minimal section of ϑ. Furthermore, if s ∈ D(ϑ), |ϑ λ (s)| ≤ |ϑ 0 (s)| for every λ > 0, and |ϑ λ (s)| is nondecreasing in λ. Given a maximal monotone graph ϑ in R × R with 0 ∈ ϑ(0), we define, for s ∈ D(ϑ), Note that the Yosida approximation (ϑ + ) λ of ϑ + is nondecreasing in λ > 0 and (ϑ − ) λ is nonincreasing in λ > 0. Observe also that (ϑ + ) λ (s) = 0 for s ≤ 0 and (ϑ − ) λ (s) = 0 for s ≥ 0, for every λ > 0, and Given a maximal monotone graph ϑ with 0 ∈ D(ϑ), j ϑ (r) := r 0 ϑ 0 (s)ds, r ∈ R, defines a convex and lower semicontinuous function such that ϑ is equal to the subdifferential of j ϑ : Moreover, if j ϑ * is the Legendre transform of j ϑ , then We now recall a Bénilan-Crandall relation between functions u, v ∈ L 1 (Ω, ν). Denote by J 0 and P 0 the following sets of functions: J 0 := {j : R → [0, +∞] : j is convex, lower semicontinuous and j(0) = 0}, Assume that ν(Ω) < +∞ and let u, v ∈ L 1 (Ω, ν). 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): Ω vρ(u)dν ≥ 0 for every ρ ∈ P 0 ⇐⇒ u ≪ u + λv for every λ > 0, 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, and which were introduced in [43].
An example of a function ap satisfying the above assumptions is where ϕ : X → R is a measurable function satisfying 0 < c ≤ ϕ ≤ C, where c and C are constants. In particular, if ϕ(x) = 2 for every x ∈ X, is the (nonlocal) p-Laplacian operator on the metric random walk space [X, d, m].
Observe that divmapu(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 ⊂ X with ν(W ) > 0. The Gunzburger-Lehoucq type Neumann boundary operator on ∂mW is given by where, taking into account the supports of the mx, we have that, in fact, the integral is being calculated over the nonlocal tubular boundary ∂mW ∪ ∂m(X \ W ) of W . On the other hand, the Dipierro-Ros-Oton-Valdinoci type Neumann boundary operator on ∂mW is given by for which, in this case, the integral is being calculated over the nonlocal boundary ∂m(X \ W ) of X \ W .
For each of these Neumann boundary operators and for ϕ defined on Wm = W ∪ ∂mW , we can look for solutions of the following problem x ∈ ∂mW, j ∈ {1, 2}. Observe that, by the reversibility of ν with respect to m and recalling the definitions of ∂mW and Wm (Definition 1.4), mx(X \ Wm) = 0 for ν-a.e. x ∈ W . Indeed, Let Ω ⊂ X be a ν-finite set and let {u k } k∈N ⊂ L p (Ω, ν) such that u k k −→ u ∈ L p (Ω, ν) in L p (Ω, ν) and pointwise ν-a.e. in Ω. Suppose also that there exists h ∈ L p (Ω, ν) such that |u k | ≤ h ν-a.e. in Ω. Then and, in particular, Ω ap(·, y, ∇u k (·, y))dm (·) (y) k −→ Ω ap(·, y, ∇u(·, y))dm (·) Taking a subsequence if necessary, the ν-a.e. pointwise convergence and the domination by the function h in the hypotheses are a consequence of the convergence in L p (Ω, ν).

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 x ∈ ∂mW, where W ⊂ X is m-connected and ν(Wm) < +∞. See [5] and [15] for the reference local models. In Subsection 2.3 we address this problem but with the nonlocal Neumann boundary operator N ap 2 instead. Problem (2.7) is a particular case (recall (2.6)) of the following general, and interesting by itself, problem.
Assumption 5 We assume that the following generalised Poincaré type inequality holds: For every 0 < l ≤ ν(Ω), there exists a constant Λ > 0 such that, for every u ∈ L p (Ω, ν) and any measurable set Z ⊂ Ω with This assumption holds true in many important examples (see Appendix A).
From now on in this subsection we work under Assumptions 1 to 5.
A subsolution (supersolution) of (GPϕ) is a pair [u, v] with u ∈ L p (Ω, ν) and v ∈ L 1 (Ω, ν) satisfying 1., 2., 3. and Remark 2.5 (Integration by parts formula) The following integration by parts formula which results from the reversibility of ν with respect to m, can be easily proved. Let u be a measurable function such that 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 ν over Ω 1 and Ω 2 , respectively, adding them and using the integration by parts formula we get Moreover, as a consequence of these computations and (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 : R → R we obtain The next result gives a maximum principle for solutions of Problem (GPϕ) given in (2.8) and, consequently, also for solutions of Problem (2.7). Theorem 2.6 (Contraction and comparison principle) Let ϕ 1 , ϕ 2 ∈ L 1 (Ω, ν). Let [u 1 , v 1 ] be a subsolution of (GPϕ 1 ) and [u 2 , v 2 ] be a supersolution of (GPϕ 2 ). Then, in Ω and u 1 − u 2 is ν-a.e. equal to a constant.

⊓ ⊔
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.
It is easy to see that A is continuous and, moreover, it is monotone and coercive in L p (Ω, ν). 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 ap has been neglected because it is nonnegative, as shown in Remark 2.5): [20,Corollary 30], there exist u n,k ∈ L p (Ω, ν), n, k ∈ N, such that A 1 (u n,k ), A 2 (u n,k ) = ϕ.

Monotonicity of the solutions of the approximate problems
Using that ϕ n,k is nondecreasing in n and nonincreasing in k, and thanks to the way in which we have approximated the maximal monotone graphs γ and β, we obtain monotonicity properties for the solutions of the approximate problems. Fix k ∈ N. Let n 1 < n 2 . Multiply equations (2.20) and (2.21) with n = n 1 by (u n1,k − u n2,k ) + , integrate with respect to ν over Ω 1 and Ω 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 (γ + ) k and (β + ) 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 (γ − )n 1 ≥ (γ − )n 2 , (β − )n 1 ≥ (β − )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, Similarly, we obtain that, for a fixed n, u n,k is ν-a.e. in Ω nonincreasing in k.

An L p -estimate for the solutions of the approximate problems
Multiplying (2.20) and (2.21) by integrating with respect to ν over Ω 1 and Ω 2 , respectively, adding both equations and using the integration by parts formula (Remark 2.5) we get

(2.24)
For the first summand on the left hand side of (2.24) we have Since F n,k (s) := 1 n |s| p−2 s + − 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 = ν(Ω 1 ) (let Λ 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 = ν(Ω 1 ) and with l = ν(Ω 2 ) (let Λ 2 denote the constant appearing in the Poincaré type inequality for the latter case), we obtain Therefore, by (2.24) and the subsequent equations, (2.25)

Existence of solutions of (GPϕ)
Observe that a solution (u, v) of (GPϕ) satisfies 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.
Observe then that any solution (u, v) of (GPϕ) under such assumptions also satisfies This will be used later on. We divide the proof into three cases.
Step A (Boundedness). Let us first see that { u n,k L p (Ω,ν) } n,k is bounded.
Step 2. Using that R − γ,β = −∞ we obtain that { u − n,k L p (Ω,ν) } n,k is bounded with an analogous argument. Consequently, we get that { u n,k L p (Ω,ν) } n,k is bounded as desired.
Step E (Boundedness of { u k L p (Ω,ν) } k and passing to the limit in k) We now see that { u k L p (Ω,ν) } k is bounded. Since u + Therefore, for each k ∈ N, either Therefore, . Now, if inf Ran(γ) + δ ′ 4ν(Ω1) ≥ 0 then z k ≥ 0 in K k , thus u − n,k = 0 in K k and u − k L p (K k ,ν) = 0; so by the generalised Poincaré type inequality and (2.25) we get that { u k L p (Ω,ν) }n is bounded. If inf Ran(γ) + δ ′ 4ν(Ω1) < 0, then and by the generalised Poincaré inequality and (2.25) we get that { u k L p (Ω,ν) } k is bounded. Similarly for k ∈ N such that (2.37) holds. Now, proceeding as in Step C of the previous proof, we finish this proof.

⊓ ⊔
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 Suppose that (2.38) holds and let ϕ ∈ L p ′ (Ω, ν) satisfying If (2.39) holds and we have ϕ ∈ L p ′ (Ω, ν) satisfying Ω ϕdν < R + γ,β , the argument is analogous. Let ϕ n,k be defined as in (2.19) and let u n,k ∈ L ∞ (Ω, ν), n, k ∈ N, be the solution of the Approximate Problem (2.20)-(2.21). Then, by Lemma A.7 together with (2.22), { u + n,k L p (Ω,ν) } n,k is bounded. However, for a fixed k ∈ N, since u n,k is nondecreasing in n, { u − n,k L p (Ω,ν) }n is also bounded. Therefore, proceeding as in Step B of the first case, we obtain u k ∈ L p (Ω, ν), for ν-a.e. x ∈ Ω 1 , and for ν-a.e. x ∈ Ω 2 ; where, for k ∈ N, in Ω 2 . We now prove that { u k L p (Ω,ν) } k is bounded. Proceeding as in Step E of the previous proof and using the same notation, we get that for each k ∈ N, either (2.42) Case 1. For k ∈ N such that (2.41) holds, let Then, .

(2.43)
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 From here we conclude as in the previous proof. Case 2. For k ∈ N such that (2.42) holds, let and proceed similarly.
where cp is the constant in (2.5), and Λ 1 and Λ 2 come from the generalised Poincaré type inequality and depend only on p, Ω 1 and Ω 2 .
(ii) Observe that, on account of (2.4) and the above estimate, we have Therefore, since [u, v] is a solution of (GP (iii) When ϕ = 0 in Ω 2 , we can easily get that v ≪ ϕ in Ω 1 .

Other boundary conditions
We can now ask for existence and uniqueness of solutions of the following problem (which was introduced in Section 2.1) 44) or, of the more general problem, x ∈ Ω 2 ⊆ ∂mW.
Recall that N ap 2 is defined as follows which involves integration with respect to ν only over W , or more specifically over ∂m(X \ 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 γ(r) = β(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) = (Ω 1 , Ω 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 below. Note that this kind of Poincaré type inequality holds, for example, for finite graphs; even if Ω 2 = ∂mW . Remark 2.9 Let Ω := Ω 1 ∪ Ω 2 . The following integration by parts formula holds: Let u be a measurable function such that 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 Ω i , i = 1, 2. For example, one could consider a problem, posed in Ω 1 ∪ Ω 2 ⊂ R N , such as the following where J i are kernels like the one in Example 1.2, and a i p are functions like the one in Subsection 2.1, i = 1, 2, 3. This could be done by obtaining a Poincaré type inequality involving 1 α0 J 0 , where J 0 is the minimum of the previous three kernels and α 0 = R N J 0 (z)dz. This idea has been used in [22] to study a homogenization problem.

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 conditions and on the other we tackle nonlinear boundary conditions. We work under the Assumptions 1 to 5 used in Subsection 2.2.

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 ∂mW when β = R × {0}. This includes, in particular, for γ = R × {0}, the problem where the dynamic evolution occurs only on the boundary: x ∈ ∂mW, 0 < t < T, 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 x ∈ Ω.

(3.2)
To solve this problem we use nonlinear semigroup theory. To this end we introduce a multivalued operator associated to Problem (3.2) that allows us to rewrite it as an abstract Cauchy problem. Observe that this operator is defined on L 1 (Ω, ν) ≡ L 1 (Ω 1 , ν) × L 1 (Ω 2 , ν) .
On account of the results given in Subsection 2.2 (Theorems 2.6 and 2.7) we have the following result.

Recall that an operator
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). is T -accretive in L 1 (Ω, ν) and satisfies the range condition With respect to the domain of such operator we can prove the following result.
Therefore, we also have that Proof It is obvious that

For the other inclusion it is enough to see that
.
n Ω ap(x, y, un(y) − un(x))dmx(y) ν-a.e. in Ω, we conclude that, on account of (2.4), The other cases follow similarly, we see two of them. Note that, since R − γ,β < R + γ,β , it is not possible to have γ = R × {0} and β = R × {0} simultaneously. For example, suppose that we have As before, the results in Subsection 2.2.4 ensure that there exist un ∈ L p (Ω, ν) and vn ∈ L p ′ (Ω, ν), n ∈ N, such that [un, vn] is a solution of GP

Now again, let
Then, as before, [ u, v] is a solution of GP . Consequently, again by the second part of the maximum principle, we obtain, as desired, that 0 ≤ un ≤ v for n large enough.
Finally, as a further example of a case which does not follow exactly with the same argument, suppose that γ := R × {0} and, for example, B − = 0, B + > 0.
As in the previous cases, there exist un ∈ L p (Ω, ν) and vn ∈ L p ′ (Ω, ν), n ∈ N, such that [un, vn] is a solution Then, [ u, v] is a solution of GP . Consequently, for n large enough, we get that 0 ≤ un ≤ v.

⊓ ⊔
In the next result we state the existence and uniqueness of solutions of Problem (3.2).
Let v and v be the mild solutions of Problem (3.2) with respective data v 0 , v 0 ∈ L 1 (Ω, ν) and f, f ∈ L 1 (0, T ; L 1 (Ω, ν)). Then If, in addition to the previous assumptions on the data, we impose that

4)
then the mild solution v belongs to W 1,1 (0, T ; L 1 (Ω, ν)) and satisfies Proof We start by proving the existence of mild solutions. For n ∈ N, consider the partition t n 0 = 0 < t n 1 < · · · < t n n−1 < t n n = T where t n i := iT /n, i = 1, . . . , n. Given ǫ > 0, there exists n ∈ N, f n i ∈ L p ′ (Ω, ν), i = 1, . . . n, and v n with v n i (x) ∈ γ(u n i (x)) for ν-a.e. x ∈ Ω 1 and v n i (x) ∈ β(u n i (x)) for ν-a.e. x ∈ Ω 2 , i = 1, . . . , n. That is, we may find the unique solution v n i of the time discretization scheme associated with (3.2): However, to apply the results in Subsection 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 ν over Ω we get so that, for the second step, we need Therefore, we recursively obtain that, for each n and each step i = 1, . . . , n, the following must be satisfied: However, taking n large enough, this holds thanks to (3.3), (3.5) and (3.6). Therefore, . . , n, is an ǫ-approximate solution of Problem (3.2) as defined in nonlinear semigroup theory. Consequently, by nonlinear semigroup theory (see [11], [ Uniqueness and the maximum principle for mild solutions is 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 ∈ L p ′ (Ω, ν), we may take v n 0 = v 0 for every n ∈ N in the previous computations and f n i ∈ L p ′ (Ω, ν), i = 1, . . . n, additionally satisfying Let us define Multiplying equation (3.7) by u n i and integrating over Ω with respect to ν we obtain (3.10) Consequently, Therefore, from (3.10) it follows that . . , n. Then, integrating this equation over ]t n i−1 , t n i ] and adding for 1 ≤ i ≤ n we get which, recalling the definitions of fn, un and vn, and integrating by parts, can be rewritten as Ω Ω ap(x, y, un(t)(y) − un(t)(x))(un(t)(y) − un(t)(x))dmx(y)dν(x)dt This, together with (2.5) and the fact that j * γ and j * β are nonnegative, yields Therefore, for any δ > 0, by (3.4) and Young's inequality, there exists C(δ) > 0 such that Consequently, Lemma A.7 and Lemma A.8 yield for some constant C 2 > 0. Similarly, we may find C 3 > 0 such that Consequently, by (3.12), choosing δ small enough, we deduce that {un}n is bounded in L p (0, T ; L p (Ω, ν)). Therefore, there exists a subsequence, which we continue to denote by {un}n, and u ∈ L p (0, T ; L p (Ω, ν)) such that un n ⇀ u weakly in L p (0, T ; L p (Ω, ν)).

⊓ ⊔
Observe that we have imposed the compatibility condition (3.3) because, for a strong solution, Example 3.5 Let W ⊂ X be a measurable set such that Wm is m-connected. Given f ∈ L 1 (∂mW, ν), we say that a function u ∈ L 1 (Wm, ν) is an ap-lifting of f to We define the Dirichlet-to-Neumann operator Da p ⊂ L 1 (∂mW, ν) × L 1 (∂mW, ν) as follows: where u is an ap-lifting of f to Wm. Then, rewriting the operator Da p as B m,γ,β ap for γ(r) = 0 and β(r) = r, r ∈ R, (Ω 1 = W and Ω 2 = ∂mW ), by the results in this subsection we have that Da p is T -accretive in L 1 (∂mW, ν) (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.
On account of the results given in Subsection 2.2 (Theorems 2.6 and 2.7) we have:  With respect to the domain of this operator we prove the following result.
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.
As in the previous case, by the results in Subsection 2.2.4, we know that, for n ∈ N, there exist un ∈ L p (Ω, ν), vn ∈ L p ′ (Ω 1 , ν) and wn ∈ L p ′ (Ω 2 , ν), such that [un, (vn, 1 n wn)] is a solution of GP and v < 0, the maximum principle yields un ≤ 0 ν-a.e. in Ω for every n ∈ N.
Therefore, in all the cases, {un}n is L ∞ (Ω, ν)-bounded from above. With a similar reasoning we obtain that, in any of these cases, {un}n is also L ∞ (Ω, ν)-bounded from below. Then, since n Ω ap(x, y, un(y) − un(x))dmx(y) in Ω 1 , we obtain that as desired.

⊓ ⊔
The following theorem gives the existence and uniqueness of solutions of Problem DP ap,γ,β f,v0 . Recall that Γ − < Γ + and B − < B + .

21)
the mild solution v belongs to W 1,1 (0, T ; L 1 (Ω 1 , ν)) and satisfies the equation The proof of this result differs, strongly at some points, from the proof of Theorem 3.4.
To apply these results we must ensure that holds for each step, but this holds true thanks to the choice of the f n i , i = 1, . . . , n. Therefore, . Consequently, by nonlinear semigroup theory ((see [11], Step 1. Suppose first that R − γ,β = −∞ and R + γ,β = +∞. In the construction of the mild solution, we now take v n 0 = v 0 (since v 0 ∈ L p ′ (Ω 1 , ν)) and the functions Multiplying both equations in (3.24) by u n i , integrating with respect to ν over Ω 1 and Ω 2 , respectively, and adding them, we obtain Then, since w n i (x) ∈ β(u n i (x)) for ν-a.e. x ∈ Ω 2 the second term on the left hand side is nonnegative and integrating by parts the third term we get Then, integrating this equation over ]t i−1 , t i ] and adding for 1 ≤ i ≤ n we get which, recalling the definitions of fn, un, vn and wn, can be rewritten as This, together with (2.5) and the fact that j * γ is nonnegative, yields Therefore, for any δ > 0, by (3.21) and Young's inequality, there exists C(δ) > 0 such that, in particular, (3.28) Observe also that, for any n ∈ N and i ∈ {1, . . . , n}, and for t ∈]t n i−1 , t n i ], Indeed, multiplying the first equation in (3.23) by 1 r T + r (u n i ) and integrating with respect to ν over Ω 1 , then multiplying the second by T n 1 r T + r (u n i ) and integrating with respect to ν over Ω 2 , adding both equations, neglecting the nonnegative term involving ap (recall Remark 2.5) and letting r ↓ 0, we get that i.e.,

Ω1
( Therefore, which is equivalent to (3.29). Now, by (3.25), if Γ + = +∞, there exists M > 0 such that Consequently, Lemma A.7 applied for A = Ω 1 , B = ∅ and α = γ, yields for every n ∈ N, every 0 ≤ t ≤ T and some constant K 2 > 0. Suppose now that Γ + < +∞. Then, by (3.29) we have that, for any n ∈ N and i ∈ {1, . . . , n}, and for for n sufficiently large. Therefore, we may apply Lemma A.8 for A = Ω 1 , B = ∅ and α = γ 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 δ small enough, we get T 0 un(t) p L p (Ω,ν) dt ≤ K 4 for every n ∈ N, for some constant K 4 > 0, that is, {un}n is bounded in L p (0, T ; L p (Ω, ν)). Therefore, there exists a subsequence, which we continue to denote by {un}n, and u ∈ L p (0, T ; L p (Ω, ν)) such that un n ⇀ u weakly in L p (0, T ; L p (Ω, ν)).
Proceeding in the same way we get that v n k,i ≤ v n k+1,i and u n k,i ≤ u n k+1,i ν-a.e.
Step 3. Finally, assume that both R − γ,β and R + γ,β are finite. We define, for k ∈ N, By the previous step we have that, for k large enough such that f + 1 k satisfies

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] and [43]). Here we generalise some of these results. Passing to a subsequence if necessary, we can assume that and gn(x) → 0 for every x ∈ A \ Ng, where Ng ⊂ A is ν-null.
On the other hand, by (A.1), we also have that Fn → 0 in L q (Q, ν ⊗ mx).