Abstract
We show that signed weak solutions to parabolic obstacle problems with porous medium-type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous.
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1 Introduction
Let \(\Omega _T := \Omega \times (0,T)\), where \(\Omega \subset \mathbb {R}^n\) is an open set and \(0<T<\infty \). In the present paper, we are concerned with the obstacle problem to partial differential equations, whose prototype is the porous medium equation (PME for short)
with a parameter \(q \in (0, \infty )\). If \(0<q<1\), the equation is degenerate and if \(q>1\), it is singular. More generally, for \(q \in (0, \infty )\) we are concerned with partial differential equations of the type
where \({\textbf{A}} :\Omega _T \times \mathbb {R}\times \mathbb {R}^n \rightarrow \mathbb {R}^n\) is a Carathéodory function, i.e., it is measurable with respect to \((x,t) \in \Omega _T\) for all \((u, \xi ) \in \mathbb {R}\times \mathbb {R}^n\) and continuous with respect to \((u, \xi ) \in \mathbb {R}\times \mathbb {R}^n\) for a.e. \((x,t) \in \Omega _T\). Moreover, we assume that \({\textbf{A}}\) satisfies the structure conditions
where \(C_o, C_1 >0\) are given constants. For the basic theory for the porous medium equation and its generalizations, we refer to the monographs [12, 15, 30,31,32].
We use a variational approach to define solutions to the obstacle problem to (1.1) with an obstacle function \(\psi \in C^0(\overline{\Omega _T})\). Heuristically, multiplying (1.1) by \(\varphi (v-u)\), where \(\varphi \) denotes a nonnegative cutoff function with \({{\,\textrm{spt}\,}}\varphi \Subset \Omega _{T}\) and v is a comparison map satisfying \(v \ge \psi \), and performing integration by parts, the variational inequality
must hold true for a solution u that is above the given obstacle \(\psi \). Since we do not assume regularity properties for u in the time direction, the first term is defined more rigorously in the next section. Existence results for variational solutions to the obstacle problem to porous medium type equations can be found in [1, 8, 27, 28].
At this stage, we will state our main result. The quantitative Hölder estimate (1.3) and thus the theorem is formulated for globally bounded local weak solutions u. However, by restricting ourselves to suitable compact subsets of \(\Omega _T\), an analogous result holds for locally bounded local weak solutions to the obstacle problem. For a compact subset \(K \Subset \Omega _T\), we denote the intrinsic, parabolic q-distance of K to the parabolic boundary \(\Gamma := (\Omega \times \{0\}) \cup (\partial \Omega \times (0,T))\) of \(\Omega _T\) by
If the obstacle function satisfies \(\psi \in C^{0; \beta , \frac{\beta }{2}}(\Omega _T)\), then also the Hölder seminorm \([\psi ]_{C^{0;\beta ,\frac{\beta }{2}};q}\) of \(\psi \) with respect to the intrinsic q-distance is finite.
Theorem 1.1
Let u be a bounded local weak solution to the obstacle problem to (1.1) with \(q \in (0,\infty )\) and structure conditions (1.2) and a Hölder continuous obstacle function \(\psi \in C^{0; \beta , \frac{\beta }{2}}(\Omega _T)\) for some \(\beta \in (0,1)\) in the sense of Definition 2.1. Then, u is locally Hölder continuous. More precisely, there exists an exponent \(\gamma = \gamma (n,q,C_o,C_1,\beta ) \in (0,\beta ]\) and a constant \(c= c(n,q,C_o,C_1,\beta )\) such that the bound
holds for any compact subset \(K \Subset \Omega _T\) and \((x_1,t_1), (x_2,t_2) \in K\).
1.1 Background
The theory of Hölder continuity for porous medium type equations is well developed in the obstacle-free case. The first proof goes back to DiBenedetto and Friedman [14], in which nonnegative solutions in the degenerate case were considered. The proof in the singular case can be found in [15] and for the treatment of signed solutions we mention [23]. For more recent developments, we refer to [4, 9, 24, 26]. However, for the obstacle problem the theory is not complete yet. Hölder continuity was proven for quasilinear problems in [29] and for problems with quadratic growth in [11]. In the case of porous medium type equations, Hölder continuity for nonnegative solutions to the obstacle problem has been treated in the recent papers [7] and [10]. The former concerns the degenerate case for the PME, and the latter the singular case for more general equations with structural conditions analogous to (1.2). However, especially the theory for signed solutions is missing, which we are addressing in this paper.
The obstacle problem is used as a standard tool in nonlinear potential theory; see, e.g., [18, 25]. For the porous medium equation, a crucial approximation scheme in this connection exploits the obstacle problem in [19]. As further applications, we mention questions related to boundary regularity addressed in [3, 21]. In this context, we also point out that there is an alternative approach to the obstacle problem, in which the solution is defined as the smallest weak supersolution lying above the given obstacle \(\psi \). This approach allows to consider fairly irregular obstacles as in [22]. For a study of the connection between these two different notions of solutions, we refer to [2].
1.2 Strategy of the proof
In our proof, we use a similar strategy as in [7] and [10] for nonnegative obstacles, which relies on a De Giorgi-type iteration argument. The idea is to construct a sequence of cylinders shrinking to a common vertex. In each of these cylinders, we consider measure theoretic alternatives, which we will call first and second alternative. Roughly speaking, in the first alternative the solution u is bounded away from its essential supremum or infimum in a large portion of the considered cylinder, where “large” means that the measure of the set where u is bounded away from its supremum or infimum is at least a certain fraction of the measure of the whole cylinder determined by the data n, q, \(C_o\), and \(C_1\). In contrast, in the second alternative u is close to the same extremum in some positive portion of the cylinder. For a precise definition, we refer to Sect. 5. In both alternatives, the solution is bounded away from one of its extrema in a quantifiable way pointwise almost everywhere in a smaller cylinder, where the distance of u to its extremum depends on the chosen portion in the second alternative. For the precise arguments, we refer to Sect. 5 in case of the first and Sect. 6 in case of the second alternative. We then show in Sect. 7 that when passing to the subsequent cylinder in the sequence of nested, shrinking cylinders mentioned above, the oscillation of the solution is reduced by a fixed amount. Finally, all cases considered in Sect. 7 are combined in Sect. 8 to deduce an oscillation decay estimate, which is in turn used to prove the quantitative Hölder estimate (1.3).
In the heart of the De Giorgi-type iteration are energy estimates for truncations of solutions, which are stated and proved in Sect. 3. In case of the second alternative, we also exploit the logarithmic estimates from Sect. 4. When deriving suitable energy or logarithmic estimates for a solution to the obstacle problem, we use a comparison map depending on the solution itself. Then, additional attention has to be paid to guarantee that it is admissible, especially that it is sufficiently regular in time and stays above the given obstacle.
While the energy estimate for truncations from above takes a similar form as in the obstacle-free case, the obstacle will play a role in the estimate for truncations from below. In the De Giorgi-type iteration argument, this will be taken into account in the upper bound for the oscillation of u. Namely, such an upper bound should be sufficiently large compared to the oscillation of the obstacle.
In order to balance the inhomogeneous scaling behavior of the PME, we will work in cylinders which respect the intrinsic geometry of the equation. In particular, we will use cylinders of the form
in which the scaling parameter \(\theta \) is comparable to \(|u|^{q-1}\). In contrast to the proof for the singular equations in [10], we treat both degenerate and singular cases with cylinders taking the same form (as in the obstacle-free case [23]).
We will separate different cases in the proof: when the solution u is near zero compared to its oscillation, and when u is bounded away from zero by a fraction of its oscillation. In the latter case, the equation behaves like a linear one. Additional challenges in the case of signed solutions are given by the fact that when u is near zero, the sign of u may change in the considered cylinder. Particularly, when applying the second alternative we use a technical argument, which is visible in Lemma 7.1 to be able to avoid the set where u becomes degenerate/singular. Furthermore, when dealing with the case where u is negative and bounded away from zero in Lemma 7.3, additional care is needed in the construction of cylinders in order to ensure that the supremum of the obstacle function is small enough compared to the solution u.
It would be interesting to obtain regularity up to the boundary when suitable boundary values are prescribed. However, this is a topic for further research.
2 Definitions and auxiliary results
In order to give a formal definition of solutions, we consider the class of functions
Admissible comparison maps will be contained in the class of functions
Definition 2.1
We say that \(u\in K_\psi (\Omega _T)\) is a local weak solution to the obstacle problem associated with (1.1) if and only if
holds true for all comparison maps \(v \in K'_\psi (\Omega _T)\) and every test function \(\varphi \in C_0^\infty (\Omega _T;\mathbb {R}_{\ge 0})\). The time term above is defined as
The proof of the following lemma follows the lines of [7, Theorem 5.1] and [10, Theorem 4.1].
Lemma 2.2
Let \(0< q < \frac{n+2}{(n-2)_+}\) and u be a local weak solution to the obstacle problem in the sense of Definition 2.1. Then, u is locally bounded.
For \(z_o = (x_o,t_o) \in \Omega _T\), we will work with cylinders of the form
The lateral boundary of \(Q_{\rho ,s} (z_o)\) is \(\partial B_{\rho }(x_o) \times (t_o-s, t_o)\) and the parabolic boundary of \(Q_{\rho ,s} (z_o)\) is \(B_{\rho }(x_o) \times \{t_o-s\} \cup \partial B_{\rho }(x_o) \times (t_o-s, t_o)\). For \(b \in \mathbb {R}\) and \(\alpha > 0\), we denote the signed \(\alpha \)-power of b by
Furthermore, we will write \({{\,\textrm{sup}\,}}\), \({{\,\textrm{inf}\,}}\), and \({{\,\textrm{osc}\,}}\) instead of \({{\,\mathrm{ess\,sup}\,}}\), \({{\,\mathrm{ess\,inf}\,}}\), and \({{\,\mathrm{ess\,osc}\,}}\) to simplify the notation.
We will exploit the following mollification in time. For \(v \in L^1(\Omega _T)\) and \(h>0\), define
We collect some useful properties of the mollification in the following lemma, see [20, Lemma 2.9] and [6, Appendix B].
Lemma 2.3
Let v and \([\![ v ]\!]_{h}\) be as above and \(p \ge 1\). Then, the following properties hold:
-
(i)
If \(v \in L^p(\Omega _T)\), then
$$\begin{aligned}{}[\![ v ]\!]_{h} \rightarrow v\quad \text { in } L^p(\Omega _T) \text { as } h \rightarrow 0. \end{aligned}$$ -
(ii)
Let \(v\in L^p(0,T; W^{1,p}(\Omega ))\). Then,
$$\begin{aligned}{}[\![ v ]\!]_{h}\rightarrow v\quad \text { in } L^p(0,T;W^{1,p}(\Omega )) \text { as } h \rightarrow 0. \end{aligned}$$ -
(iii)
If \(v \in C^0(\overline{\Omega _T})\) and \(\Omega \subset \mathbb {R}^n\) is a bounded set, then
$$\begin{aligned}{}[\![ v ]\!]_{h} \rightarrow v \quad \text { uniformly in }\Omega _T\ \text { as }\ h \rightarrow 0. \end{aligned}$$ -
(iv)
The weak time derivative \(\partial _t [\![ v ]\!]_{h}\) exists in \(\Omega _T\) and is given by the formula
$$\begin{aligned} \partial _t [\![ v ]\!]_{h} = \frac{1}{h} ( v - [\![ v ]\!]_{h} ). \end{aligned}$$
In this section, we recall some standard results needed in the proofs. We begin with a special case of the Sobolev inequality, cf. [13, Chapter I, Proposition 3.1].
Lemma 2.4
Let \(v \in L^2(0,T; W^{1,2}_0(\Omega ))\). Then, there exists \(c = c(n) > 0\) such that
We also make use of De Giorgi’s isoperimetric inequality and the so-called fast geometric convergence [13, Chapter I, Lemma 2.2], [17, Lemma 7.1], which we state next.
Lemma 2.5
Let \(k<l\) be real numbers and \(B_\rho (x_o) \subset \mathbb {R}^n\). Then, for any \(v\in W^{1,1}(B_\rho (x_o))\) there exists a constant \(c = c(n) > 0\) such that
Lemma 2.6
Suppose that \(\{Y_i\}_{i\in \mathbb {N}_0}\) is a sequence of positive real numbers that satisfy
with constants \(C,\sigma > 0\) and \(B>1\). Then, \(Y_i \rightarrow 0\) as \(i \rightarrow \infty \) whenever
3 Energy estimates
For \(w,k \in \mathbb {R}\), let us define
The following estimates follow from the definition above; see, e.g., [5, Lemma 2.2].
Lemma 3.1
There exists a constant \(c = c(q) > 0\) such that for all \(w,k \in \mathbb {R}\) and \(q > 0\), the inequality
holds true.
Next, we give energy estimates for weak solutions to the obstacle problems. Note that in the estimate involving \((u-k)_+\) only levels k larger than the obstacle function are admissible, whereas there is no restriction on the admissible levels in the estimate involving \((u-k)_-\).
Lemma 3.2
Let \(z_o = (x_o,t_o) \in \Omega _T\) and \(Q_{\rho ,s}(z_o) \Subset \Omega _T\). Further, for \(\psi \in C^0(\Omega _T)\) we assume that \(u \in K_\psi (\Omega _T)\) is a local weak solution to the obstacle problem to (1.1) with structure conditions (1.2) in the sense of Definition 2.1. Then, for any function \(\varphi \in C^\infty (Q_{\rho ,s}(z_o);\mathbb {R}_{\ge 0})\) vanishing on the lateral boundary of \(Q_{\rho ,s}(z_o)\) there exists \(c = c(C_o,C_1) > 0\) such that the following estimates hold.
-
(i)
For all \(k \ge \sup _{Q_{\rho ,s}(z_o)} \psi \), we have
$$\begin{aligned} \max \bigg \{&{{\,\textrm{sup}\,}}_{t_o -s< t < t_o} \int _{B_\rho (x_o) \times \{t\}} \varphi ^2 {\mathfrak {g}}_{+}(u,k) \, \textrm{d}x, \tfrac{C_o}{2} \iint _{Q_{\rho ,s}(z_o)} \varphi ^2 |\nabla (u-k)_{+}|^2 \, \textrm{d}x \textrm{d}t \bigg \} \\&\le c \iint _{Q_{\rho ,s}(z_o)} [(u-k)_{+}^2|\nabla \varphi |^2 + {\mathfrak {g}}_{+} (u,k) |\partial _t \varphi ^2|] \, \textrm{d}x \textrm{d}t \\&\quad + \int _{B_\rho (x_o) \times \{t_o -s\}} \varphi ^2 {\mathfrak {g}}_{+}(u,k)\, \textrm{d}x. \end{aligned}$$ -
(ii)
For arbitrary \(k \in \mathbb {R}\), we have
$$\begin{aligned} \max \bigg \{&{{\,\textrm{sup}\,}}_{t_o -s< t < t_o} \int _{B_\rho (x_o) \times \{t\}} \varphi ^2 {\mathfrak {g}}_{-}(u,k)\, \textrm{d}x, \tfrac{C_o}{2} \iint _{Q_{\rho ,s}(z_o)} \varphi ^2 |\nabla (u-k)_{-}|^2 \, \textrm{d}x \textrm{d}t \bigg \} \\&\le c \iint _{Q_{\rho ,s}(z_o)} [(u-k)_{-}^2|\nabla \varphi |^2 + {\mathfrak {g}}_{-} (u,k) |\partial _t \varphi ^2|] \, \textrm{d}x \textrm{d}t \\&\quad + \int _{B_\rho (x_o) \times \{t_o -s\}} \varphi ^2 {\mathfrak {g}}_{-}(u,k) \, \textrm{d}x. \end{aligned}$$
Proof
In the following, we omit \(z_o\) for simplification. We first prove (i). Let \(\varphi \in C^\infty (Q_{\rho ,s};\mathbb {R}_{\ge 0})\) vanish on the lateral boundary \(\partial B_\rho \times (-s,0)\) of \(Q_{\rho ,s}\). Further, we define \(\xi _\varepsilon \in W_0^{1,\infty }([-s,0];[0,1])\) by
and use \(\eta := \varphi ^2 (\xi _\varepsilon )_\delta \), in which \((\xi _\varepsilon )_\delta \) is a standard mollification [17, Sect. 2.2] of \(\xi _\varepsilon \) with \(0<\delta < \tfrac{\varepsilon }{2}\), as a test function in (2.1). Moreover, we define
Note that there hold \(w_h \in C^0((0,T);L^{q+1}_{{{\,\textrm{loc}\,}}}(\Omega )) \cap L^2_{{{\,\textrm{loc}\,}}}(0,T;H^1_{{{\,\textrm{loc}\,}}}(\Omega ))\) and
by Lemma 2.3. Further, we have that \(w_h \ge \psi \) in \(Q_{\rho , s}\), since \([\![ u ]\!]_{h} \ge [\![ \psi ]\!]_{h}\) in \(\Omega _T\) and \(k \ge \sup _{Q_{\rho ,s}} \psi \). Since it is sufficient that \(w_h\) satisfies the obstacle condition in \({{\,\textrm{spt}\,}}(\eta ) \subset Q_{\rho ,s}\), \(w_h\) is an admissible comparison map in (2.1). Therefore, we obtain that
In the following, we treat the two terms separately. First, by the formula for \(\partial _t w_h\) above and since \(\big ( \varvec{u}^q - \varvec{[\![ u ]\!]_{h}^q} \big ) \big ( u - [\![ u ]\!]_{h} \big ) \ge 0\), we conclude that
Integrating by parts leads to
Writing the last term on the right-hand side of the preceding inequality as
yields
Inserting this into the first term of (3.1), we see that
Hence, passing to the limit \(h \downarrow 0\) with the aid of Lemma 2.3 (i) and (iii), we obtain that
In order to treat the second term in (3.1), observe that
by Lemma 2.3 (ii). Since \({\textbf{A}}(x,t,u,\nabla u) \in L^2(Q_{\rho ,s},\mathbb {R}^n)\) by growth condition (1.2)\(_2\), this implies
By means of (1.2)\(_1\) and since \(\eta \ge 0\), we observe that
Further, by (1.2)\(_2\) and Young’s inequality with parameter \(\frac{C_o}{4 C_1}\) (see [16, Appendix B]), we find that
Inserting the preceding two inequalities into (3.3), we obtain that
Together with (3.1) and (3.2), we conclude that
By first passing to the limit \(\delta \downarrow 0\), and subsequently \(\varepsilon \downarrow 0\), we get
Since all terms are nonnegative, we infer the desired energy estimate by first discarding the first term on the left-hand side and passing to the limits \(t_1 \downarrow -s\), \(t_2 \uparrow 0\) and then discarding the second term on the left-hand side, passing to the limit \(t_1 \downarrow -s\) and taking the supremum over all \(t_2 \in (-s,0)\).
In order to prove (ii), we use the comparison function \(w_h := [\![ u ]\!]_{h} + ([\![ u ]\!]_{h} - k)_- + \Vert \psi - [\![ \psi ]\!]_{h}\Vert _{L^\infty (Q_{\rho ,s})}\) with an arbitrary level \(k \in \mathbb {R}\) and proceed similarly as in (i). \(\square \)
4 Logarithmic estimates
In this section, we will state a logarithmic estimate as in [7] and [10]. Let \(0< \gamma < \Gamma \) and define
Observe that for \(a \le \Gamma \), we have
with \(\phi (a) = 0\) for \(a \le \gamma \). Further, we have \(\phi '' (a) = \left( \phi ' \right) ^2(a)\) for \(a \ne \gamma \). Note that \(\phi ^2\) is differentiable in \([0,\Gamma ]\) such that the Lipschitz continuous derivative \(\left( \phi ^2 \right) '\) satisfies
With this information at hand, we are able to prove the following lemma.
Lemma 4.1
Let \(B_{\rho _1}(x_o) \Subset B_{\rho _2}(x_o) \Subset \Omega \), \(0< t_1< t_2 < T\) and \(Q_2 := B_{\rho _2}(x_o) \times (t_1,t_2)\). Further, define \(\Gamma := {{\,\textrm{sup}\,}}_{Q_2} (u-k)_+ \) and consider some parameter \(\gamma \in (0,\Gamma )\). Assume that \(\psi \in C^0(\Omega _T)\) and let \(u \in K_\psi (\Omega _T)\) be a local weak solution to the obstacle problem (1.1) with structure conditions (1.2) according to Definition 2.1. Then, there exists \(c = c(q,C_o,C_1) > 0\) such that the following estimates hold.
-
(i)
For any \(k \ge \sup _{Q_2} \psi \), we have
$$\begin{aligned} \sup _{t \in (t_1,t_2)} \int _{B_{\rho _1} (x_o)} \int _k^u&|s|^{q-1} \left( \phi ^2 \right) '((s - k)_+) \, \textrm{d}s \textrm{d}x \\&\le \int _{B_{\rho _2}(x_o) \times \{t_1\}} \int _k^u |s|^{q-1} \left( \phi ^2 \right) '((s-k)_+) \, \textrm{d}s \textrm{d}x \\&\quad + \frac{c}{(\rho _2 - \rho _1)^2} \iint _{Q_2} \phi \left( (u-k)_+ \right) \, \textrm{d}x \textrm{d}t. \end{aligned}$$ -
(ii)
For any \(k \in \mathbb {R}\), we have that
$$\begin{aligned} \sup _{t \in (t_1,t_2)} \int _{B_{\rho _1} (x_o)} \int _u^k&|s|^{q-1} \left( \phi ^2 \right) '((s - k)_-) \, \textrm{d}s \textrm{d}x \\&\le \int _{B_{\rho _2}(x_o) \times \{t_1\}} \int _u^k |s|^{q-1} \left( \phi ^2 \right) '((s-k)_-) \, \textrm{d}s \textrm{d}x \\&\quad + \frac{c}{(\rho _2 - \rho _1)^2} \iint _{Q_2} \phi \left( (u-k)_- \right) \, \textrm{d}x \textrm{d}t. \end{aligned}$$
Proof
In the following, we omit \((x_o,t_o)\) for simplicity. We start with the proof of (i). Since all terms in the asserted estimate depend continuously on k, we may assume that \(k > \sup _{Q_2} \psi \). We would like to use
with
as comparison map in (2.1). By Lemma 2.3 and since \(\big ( \phi ^2 \big )'\) is a Lipschitz continuous function, we have that \(w_h \in C^0((0,T); L^{q+1}_{{{\,\textrm{loc}\,}}}(\Omega )) \cap L^2_{{{\,\textrm{loc}\,}}}(0,T; H^1_{{{\,\textrm{loc}\,}}}(\Omega ))\) with \(\partial _t w_h \in L^{q+1}_{{{\,\textrm{loc}\,}}}(\Omega _T)\). Moreover, if \([\![ u ]\!]_{h} \le k\), we find that
in \(Q_2\), and for \([\![ u ]\!]_{h} > k\) we have that
by the restriction on \(\lambda \). Consequently, \(w_h\) is an admissible comparison map in (2.1). Thus, for any \(\varphi \in C^\infty _0(Q_2; \mathbb {R}_{\ge 0})\) we obtain that
In the following, we estimate these terms separately. First, we calculate
Since the derivative \(\partial _t [\![ u ]\!]_{h}\) vanishes a.e. in the set \(\big \{ \big ( [\![ u ]\!]_{h}-k \big )_+ = \gamma \big \}\), the terms involving \(\big ( \phi ^2 \big )''\) are well defined a.e. in \(Q_2\). Further, decreasing \(\lambda \) if necessary, we may assume that the last factor is positive, which allows us to estimate
by recalling \(\big ( \varvec{u}^q - \varvec{[\![ u ]\!]_{h}^q} \big ) \big ( u - [\![ u ]\!]_{h} \big ) \ge 0\). Together with integration by parts and the fact that
this implies that
The last term is a result of integration by parts once more. Recalling the definition of \(\textrm{I}_h\) and inserting the preceding inequality yields
Therefore, passing to the limit \(h \downarrow 0\), by means of Lemma 2.3 we obtain that
Next, we turn our attention to \(\textrm{II}_h\). By Lemma 2.3 and since \((\phi ^2)'\) is Lipschitz continuous, we find that
Together with the structure conditions (1.2), this implies that
Here, the term involving \(\big ( \phi ^2 \big )''\) is well defined a.e. in \(\Omega _T\), since \(\nabla (u-k)_+ = 0\) a.e. in \(\{ (u-k)_+ = \gamma \}\). At this point, we choose \(\varphi (x,t) = \xi _\varepsilon (t) \eta (x)^2\), where \(\xi _\varepsilon \) is defined as in the proof of Lemma 3.2 and \(\eta \in C_0^1(B_{\rho _2},\mathbb {R}_{\ge 0})\) is a cutoff function with \(\eta = 1\) in \(B_{\rho _1}\) and \(|\nabla \eta | \le \frac{2}{\rho _2-\rho _1}\). Applying Young’s inequality with parameter \(\frac{C_o}{2C_1}\) (see [16, Appendix B]) yields
In the last line, we used that \(2\phi \left( \phi '\right) ^2 - \left( \phi ^2 \right) '' = - 2 \left( \phi ' \right) ^2 \le 0\). Together with (4.1), we obtain that
Passing to the limit \(\varepsilon \downarrow 0\), we conclude that
for any \(t \in (t_1,t_2)\), which proves (i).
For the case (ii), we start with the comparison function
since \(\phi , \phi ' \ge 0\) and proceed similarly as in the case (i). \(\square \)
5 First alternative
In the following, we use parameters \(\varvec{\mu }^-, \varvec{\mu }^+ \in \mathbb {R}\) and \(\varvec{\omega } >0\) satisfying
where slightly different factors \(\theta \approx |u|^{q-1}\) will be considered. We will distinguish between the case where u is close to the value zero in \(Q_{\rho , \theta \rho ^2}(z_o)\),
the case where u is positive and bounded away from zero in \(Q_{\rho , \theta \rho ^2}(z_o)\),
and the case where u is negative and bounded away from zero in \(Q_{\rho , \theta \rho ^2}(z_o)\),
While the degeneracy/singularity of (1.1) has to be taken into account in the first case, (1.1) does not become degenerate/singular in \(Q_{\rho , \theta \rho ^2}(z_o)\) in the latter two cases.
By using (5.1)\(_3\), we can write (5.2) equivalently as
which corresponds to the region \(\textrm{I} = \textrm{I}_1 \cup \textrm{I}_2\) in Fig. 1. We split \(\textrm{I}\) into subcases \(\textrm{I}_1\) and \(\textrm{I}_2\) by observing that by (5.1)\(_3\) one of the cases
must hold, which is equivalent to
Here, (5.5) and (5.6)\(_1\) correspond to region \(\textrm{I}_1\), while (5.5) and (5.6)\(_2\) correspond to \(\textrm{I}_2\) in Fig. 1. Furthermore, condition (5.3) is equivalent to
which corresponds to region \(\textrm{II}\) in Fig. 1 and (5.4) is equivalent to
corresponding to region \(\textrm{III}\) in Fig. 1.
Relevant ranges for the parameters \(\varvec{\mu }^-\) and \(\varvec{\mu }^+\)
In each of the cases (5.2) – (5.4), for some constant \(\nu \in (0,1)\) we will be concerned with the measure theoretic alternatives
or with the measure theoretic alternatives
In the so-called first alternative (5.7)\(_1\)/(5.8)\(_1\), the solution u is bounded away from its essential supremum or infimum on a large portion of the considered cylinder, whereas it is close to the essential supremum or infimum on a large part of the cylinder in the second alternative (5.7)\(_2\)/(5.8)\(_2\). In both situations, our goal is to show that u is bounded away from one of the extreme values \(\varvec{\mu }^+/\varvec{\mu }^-\) pointwise a.e. in a suitable sub-cylinder of \(Q_{\rho , \theta \rho ^2}(z_o)\) together with a quantitative bound. The necessary tools for the first alternatives (5.7)\(_1\)/(5.8)\(_1\) will be discussed in the present section, while we will be concerned with the second alternatives (5.7)\(_2\)/(5.8)\(_2\) in Sect. 6. More precisely,
-
for case \(\textrm{I}_1\) we use the alternatives (5.7) together with Lemmas 5.3 and 6.3;
-
for case \(\textrm{I}_2\) we use the alternatives (5.8) together with Lemmas 5.4 and 6.4;
-
for case \(\textrm{II}\) we use the alternatives (5.8) together with Lemmas 5.2 and 6.4;
-
for case \(\textrm{III}\) we use the alternatives (5.7) together with Lemmas 5.1 and 6.3.
We start with the non-degenerate/non-singular cases (5.3) and (5.4), since the corresponding proofs are slightly easier as in the case (5.2).
5.1 De Giorgi-type lemmas in the non-degenerate/non-singular case
In this section, we state De Giorgi-type lemmas for the case where u is bounded away from its supremum and the case where u is bounded away from its infimum. However, we only prove the former one, since the proof of the latter is analogous. For the first lemma, we suppose that (5.4) holds true. Further, in the first lemma we use the scaling
Lemma 5.1
Assume that (5.4) and (5.9) hold and let u be a locally bounded, local weak solution to the obstacle problem and \(Q_{\rho ,\theta \rho ^2}(z_o) \Subset \Omega _T\). Furthermore, suppose that \(\tfrac{1}{2} \left( \varvec{\mu }^+ + \varvec{\mu }^- \right) \ge \sup _{Q_{\rho ,\theta \rho ^2}(z_o)} \psi \). Then, there exists a constant \(\nu = \nu (n,q,C_o,C_1) \in (0,1)\) such that if
then
Proof
For simplicity, we omit the reference point \(z_o\) in the notation. Observe that
up to a constant depending only on q by Lemma 3.1. From the energy estimate, Lemma 3.2 (i), we obtain
for \(k \ge \sup _{Q_{\rho ,\theta \rho ^2}} \psi \) and \(\varphi \in C^\infty (Q_{\rho ,\theta \rho ^2}; \mathbb {R}_{\ge 0})\) vanishing on the parabolic boundary of \(Q_{\rho ,\theta \rho ^2}\). For \(j \in \mathbb {N}_0\), we choose
Note that \(k_j \ge k_0 = \frac{1}{2} (\varvec{\mu }^+ + \varvec{\mu }^-) \ge \sup _{Q_0} \psi \ge \sup _{Q_j} \psi \) for any \(j \in \mathbb {N}_0\), since \(Q_{\rho ,\theta \rho ^2} = Q_0 \supset Q_1 \supset \ldots \) and by the assumption on \(k_0\). Furthermore, we use a smooth cutoff function \(0 \le \varphi \le 1\) vanishing on the parabolic boundary of \(Q_j\) and equal to 1 in \({Q}_{j+1}\) such that
Observe that (5.4) in particular implies that \(k_j< \varvec{\mu }^+ < 0\) for any \(j \in \mathbb {N}_0\). Thus, we know that \(2 \left| \varvec{\mu }^+ \right| \le 2|u| \le |u|+ |k_j| \le 2 |k_j|\) in \(A_j := \{ u > k_j \} \cap Q_j\), which gives us that
Observe that the bounds \(\frac{1}{5} |\varvec{\mu }^-|< |\varvec{\mu }^+| < |\varvec{\mu }^-|\) and \(|\varvec{\mu }^+|< |k_j| < |\varvec{\mu }^-|\) hold true by (5.4) and the definition of \(k_j\). Taking also (5.11) into account, we estimate the preceding inequality further. In particular, we conclude that
Next, note that \(u - k_j \ge {k}_{j+1} - k_j = 2^{-(j+3)} \varvec{\omega }\) if \(u \ge {k}_{j+1}\). By Hölder’s inequality and Lemma 2.4, we find that
In the penultimate line, we used the energy estimate and (5.10). This implies that
Dividing the preceding inequality by \(|Q_{j+1}|\) and denoting \(Y_j = |A_j|/|Q_j|\), we have that
Thus, we are able to conclude the proof by using Lemma 2.6. \(\square \)
Next, we suppose that (5.3) holds true. Further, we use the scaling
Lemma 5.2
Consider \(Q_{\rho ,\theta \rho ^2} \Subset \Omega _T\), assume that (5.3) and (5.9) hold and let u be a locally bounded, local weak solution to the obstacle problem. Then, there exists a constant \(\nu = \nu (n,q,C_o,C_1) \in (0,1)\) such that if
then
5.2 De Giorgi-type lemmas in the degenerate/singular case
Next, we will state and prove a De Giorgi-type lemma in the case where \(|\varvec{\mu }^+| \le \frac{5}{4} \varvec{\omega }\) and u is bounded away from its supremum in a significant portion of the considered intrinsic cylinder. Note that the condition \(|\varvec{\mu }^+| \le \frac{5}{4} \varvec{\omega }\) is in particular implied by (5.2).
Lemma 5.3
Let u be a locally bounded, local weak solution to the obstacle problem and \(Q_{\rho ,\theta \rho ^2} (z_o) \Subset \Omega _T\), where \(\theta = \varvec{\omega }^{q-1}\). Furthermore, assume that \(|\varvec{\mu }^+| \le \frac{5}{4} \varvec{\omega }\) and that \(\tfrac{1}{2} (\varvec{\mu }^+ + \varvec{\mu }^-) \ge \sup _{Q_{\rho ,\theta \rho ^2}(z_o)} \psi \). Then, there exists a constant \(\nu = \nu (n,q,C_o,C_1) \in (0,1)\), such that if
then
Proof
We omit the fixed reference point \(z_o\) for simplicity. By Lemma 3.1, there holds
for any \(k \in \mathbb {R}\). Further, for \({\tilde{k}} > k\) we have \((u-k)_+ \ge (u- {\tilde{k}})_+ \). From the energy estimate, Lemma 3.2 (i), we obtain
for any \({\tilde{k}} > k \ge \sup _{Q_{\rho ,\theta \rho ^2}} \psi \) and \(\varphi \in C^\infty (Q_{\rho ,\theta \rho ^2}; \mathbb {R}_{\ge 0})\) vanishing on the parabolic boundary of \(Q_{\rho ,\theta \rho ^2}\). For \(j \in \mathbb {N}_0\), we define \(k_j\), \(\rho _j\), \(B_j\), and \(Q_j\) as in Lemma 5.1, and furthermore
Let \(0 \le \varphi \le 1\) such that \(\varphi \) vanishes on the parabolic boundary of \(Q_j\), equals to 1 in \({\widetilde{Q}}_j\) and satisfies
Moreover, we set \(A_j = \left\{ u > k_j \right\} \cap Q_j\).
By formulating the preceding energy estimate for these quantities and using the assumption \(|\varvec{\mu }^+| \le \frac{5}{4} \varvec{\omega }\), we conclude that
on the set where \(u \ge {\tilde{k}}_j\). By recalling that \(\theta = \varvec{\omega }^{q-1}\), we find that
In particular, the estimate above gives us that
holds true. By introducing a smooth cutoff function \(0 \le \phi \le 1\), such that \(\phi \) equals 1 in \(Q_{j+1}\) and vanishes on the lateral boundary of \({\widetilde{Q}}_j\) with \(|\nabla \phi |\le c 2^j \rho ^{-1}\), Hölder’s inequality and Lemma 2.4 imply
In the penultimate line, we used the energy estimate and (5.12). This implies that
By dividing this by \(|Q_{j+1}|\) and denoting \(Y_j = |A_j|/|Q_j|\), we have
Then, if \(\nu \le c^{-(n+2)} B^{-(n+2)^2}\), where \(B = 2^{2 + \frac{(q-1)_+}{n+2}}\) we may use Lemma 2.6 to conclude the proof. \(\square \)
We state the De Giorgi-type lemma for the case where \(|\varvec{\mu }^-| \le \frac{5}{4} \varvec{\omega }\) (which holds in particular if (5.2) is satisfied) and u is away from its infimum without proof. However, it can be proven analogous to Lemma 5.3 by defining \(k_j = \varvec{\mu }^- + \tfrac{\varvec{\omega }}{4} + \tfrac{\varvec{\omega }}{2^{j+2}}\) and exploiting the energy estimate in Lemma 3.2 (ii). Observe that in this case any level is admissible in the energy estimate.
Lemma 5.4
Let u be a locally bounded, local weak solution to the obstacle problem and \(Q_{\rho ,\theta \rho ^2}(z_o) \Subset \Omega _T\) with \(\theta = \varvec{\omega }^{q-1}\) and assume that \(|\varvec{\mu }^-| \le \frac{5}{4} \varvec{\omega }\). Then, there exists a constant \(\nu = \nu (n,q,C_o,C_1) \in (0,1)\) such that if
then
6 Second alternative
6.1 Second alternative near infimum
Suppose that \(\varvec{\mu }^+, \varvec{\mu }^-\), and \(\varvec{\omega }\) are given by (5.1). In this section, we are concerned with a subcase of (5.2) and the case (5.4). More precisely, we assume that either
holds true (which corresponds to region \(\textrm{I}_1\) in Fig. 1), or
(which corresponds to region \(\textrm{III}\) in 1). Since \(\varvec{\mu }^+ = \varvec{\mu }^- + \varvec{\omega }\), (6.1)\(_1\) is equivalent to
Further, observe that (6.2)\(_1\) implies
First, we prove an auxiliary lemma.
Lemma 6.1
Let \(\eta \in (0,\tfrac{1}{8}]\) and \(k = \varvec{\mu }^- + \eta \varvec{\omega }\). Then, there exists \(c= c(q) > 0\) such that in case (6.10) there holds
and in the case (6.11) there holds
Furthermore, we have
Proof
First consider the case (6.1). The first estimate is a direct consequence of the definition. For the second inequality, we may estimate
which implies the claim. In case (6.2), we have
which concludes the proof of the inequalities for \(\theta \). For the last inequality, consider first (6.10) to obtain
which implies the desired inequality. In case (6.11), we may estimate
which concludes the proof. \(\square \)
Lemma 6.2
Let \(Q_{\rho ,\theta \rho ^2}(z_o) \Subset \Omega _T\) be a parabolic cylinder and \(\nu \in (0,1)\) and \(\eta \in \big (0,\frac{1}{8}\big ]\). Assume that (6.1) or (6.2) holds and that u is a locally bounded, local weak solution to the obstacle problem. Then, there exists \(\nu _1 = \nu _1(n,q,C_o,C_1,\nu ) \in (0,1)\) such that if
then
Proof
We omit \(z_o\) for simplicity and start the proof by defining
Observe that by (6.10) and (6.11) it follows that \(k_j < 0\) for all \(j \in \mathbb {N}_0\). Let \(0 \le \varphi \le 1\) be a cutoff function that equals 1 in \(Q_{j+1}\) and vanishes on the parabolic boundary of \(Q_j\) such that
From the fact that \(\varvec{\mu }^- \le u< k_j < 0\) in the set \(A_j := \{ u < k_j \} \cap Q_j\), Lemma 3.1 and the energy estimate in Lemma 3.2 (ii), we then obtain
with a constant \(c = c(n,q,C_o,C_1,\nu )\), where we used Lemma 6.1 in the last line. With these estimates at hand and Lemma 6.1, we infer in particular
Since \(k_j - u \ge k_j - k_{j+1} = 2^{-(j+2)} \eta \varvec{\omega }\) in the set \(\{u \le k_{j+1}\}\), by Hölder’s inequality and Lemma 2.4 we obtain that
Dividing by \(|Q_{j+1}|\) and denoting \(Y_j = |A_j| / |Q_j|\), we conclude that
for a constant \(c = c(n,q,C_o,C_1,\nu )\). Setting \(\nu _1 \le c^{-(n+2)} 4^{-(n+2)^2}\), we conclude the proof by using Lemma 2.6. \(\square \)
At this stage, we state the main result in this section, which allows us to deal with arbitrary \(\nu \) in the assumed measure estimate. In contrast, in the preceding lemma \(\nu _1\) is a fixed constant depending only on the data.
Lemma 6.3
Let \(Q_{2\rho ,\theta (2\rho )^2}(z_o) \Subset \Omega _T\) be a parabolic cylinder. Assume that (6.1) or (6.2) holds and that u is a locally bounded, local weak solution to the obstacle problem. Then, for any \(\nu \in (0,1)\) there exists a constant \(a = a(n,q,C_o,C_1,\nu ) \in \big (0,\frac{1}{64}\big ]\) such that if
then
Proof
In the following, we omit \(z_o\) for simplicity. Observe that from the assumption it follows that
which further implies
for some \(t_1 \in [- \theta \rho ^2, - \frac{1}{2} \nu \theta \rho ^2]\). If this did not hold, we would have
which contradicts (6.3). We divide the rest of the proof in three steps.
In Step 1, we show that the measure information in (6.4) can be propagated to the whole interval \((t_1,0)\) by using logarithmic estimate, Lemma 4.1 (ii). More precisely, we show that there exists \(s_o\in \mathbb {N}_{\ge 5}\) such that for all \(s \ge s_o\)
holds true.
In Step 2, we show that the measure estimate holds in a parabolic cylinder. More precisely, for parameter \(\nu _1 \in (0,1)\) from Lemma 6.2, we show that there exists \(s_1 \in \mathbb {N}_{\ge 2}\) such that
In Step 3, we use (6.6) together with Lemma 6.2 to conclude the result.
Step 1. Let us define the level k by
where \(\delta \in \big (0,\frac{1}{8} \big ]\). This implies
by the bound (6.10)\(_1\) or (6.11)\(_1\) depending on the case. Let us choose \(s_o \in \mathbb {N}_{\ge 5}\) large enough such that
which implies that
for every \(s \ge s_o\). First, let us suppose that
Then, we have that
for all \(t \in (t_1,0)\), which implies (6.5) since \(\tfrac{\delta }{2} > \tfrac{1}{2^s}\) for \(s \ge s_o\). If (6.7) does not hold, then we have
and let us define
Now, it follows that \( \tfrac{\delta }{2} \varvec{\omega } \le H \le \delta \varvec{\omega } \), which implies
Further, we define the function
for \(v < H + \tfrac{1}{2^s} \varvec{\omega }\). Now, we rewrite the integrals in Lemma 4.1 (ii) as
and take into account that \(|k| \le |k - \tau | \le |u|\). Thus, we deduce that
for any \(t \in (t_1,0)\) and \(\sigma \in (0,1)\). Since \(\phi \) is increasing and by (6.9), we also find that
which together with Lemma 6.1 implies
for any \(t\in (t_1,0)\), where we used also the fact that \(t_1 \ge -\theta \rho ^2\). On the left-hand side, let us consider the set \(B_{\sigma \rho } \cap \{ u(\cdot ,t) \le \varvec{\mu }^- + \tfrac{1}{2^s}\varvec{\omega } \}\) for \(t \in (t_1,0)\), where
Since the function \(\phi ( (u-k)_- )\) is decreasing in H and \(H \le \delta \varvec{\omega }\), this implies
Therefore, we find that
By combining the preceding estimates, we obtain that
Using \(\left| B_\rho \setminus B_{\sigma \rho } \right| \le n (1-\sigma ) |B_\rho |\), this yields
Let us fix
and \(\delta \) such that
This implies that
and we obtain that
for all \(t \in (t_1,0)\). Let \(s_o \in \mathbb {N}_{\ge 5}\) (depending on n, q, \(C_o\), \(C_1\) and \(\nu \)) be so large that
and
Now, we conclude that
holds for all \(t \in (t_1,0)\) and \(s \ge s_o\) or equivalently that (6.5) holds.
Step 2. To this end, we define cylinders \(Q_2= B_\rho \times \left( - \frac{1}{2} \nu \theta \rho ^2, 0 \right] \) and \(Q_1 = B_\rho \times \left( - \nu \theta \rho ^2,0\right] \), which implies that \(Q_2 \subset Q_1 \subset Q_{2\rho ,\theta (2\rho )^2}\). Further, we consider levels
and set
for \(j \in \mathbb {N}_{\ge s_o}\). By De Giorgi’s isoperimetric inequality from Lemma 2.5, we have that
Integrating over \((- \tfrac{1}{2} \nu \theta \rho ^2, 0)\), we find that
Applying Lemma 3.2 (ii), we estimate the integral on the right-hand side by
cf. Lemma 6.2. In the last line, we used Lemma 6.1. Combining the two estimates above and using \(k_j - k_{j+1} = 2^{-(j+1)} \varvec{\omega }\), we infer
At this stage, we sum over \(j = s_o, \ldots , s_o + s_1 -1\) for some \(s_1 \in \mathbb {N}_{\ge 2}\), which gives us
Choosing \(s_1\) large enough, the estimate (6.6) holds true.
Step 3. Now an application of Lemma 6.2 with \(\eta = 2^{-(s_o+s_1)}\) yields
By denoting \(a = a(n,q,C_o, C_1,\nu ) = \frac{1}{2^{s_o+s_1+1}}\), the proof is completed. \(\square \)
6.2 Second alternative near supremum
Here, we consider a subcase of (5.2) and the case (5.3). More precisely, we assume that either
holds true (which corresponds to region \(\textrm{I}_2\) in Fig. 1), or
(which corresponds to region \(\textrm{II}\) in Fig. 1). Note that (6.1)\(_1\) is equivalent to
Lemma 6.4
Let \(Q_{2\rho ,\theta (2\rho )^2}(z_o) \Subset \Omega _T\) be a parabolic cylinder such that
Assume that hypothesis (6.10) or (6.2) holds and that u is a locally bounded, local weak solution to the obstacle problem. Then, for any \(\nu \in (0,1)\) there exists a constant \(a = a(n,q,C_o,C_1,\nu ) \in \big (0,\frac{1}{64}\big ]\) such that if
then
7 Reduction in oscillation
Throughout the rest of the paper, we denote the minimum of the parameters \(\nu \) from Lemmas 5.1 to 5.4 by \(\nu _o\). Further, we let a be the minimum of the respective parameters in Lemmas 6.3 and 6.4 corresponding to the parameter \(\nu _o\) chosen above, and define \(\delta = 1-a \in [\tfrac{3}{4},1)\). This implies that these parameters coincide in the following lemmas, which allows us to use them subsequently in Sect. 8.
Moreover, throughout this section, we consider parabolic cylinders of the form \(Q_o := Q_{\rho _o, \theta \rho _o^2}(z_o)\) and quantities
Further, we assume that
We treat the following cases corresponding to (5.2)–(5.4) with \(\varvec{\mu }^\pm \), \(\varvec{\omega }\) replaced by \(\varvec{\mu }^\pm _o\), \(\varvec{\omega }_o\) separately: Either we assume that
or
or
First, we are concerned with the case where u is near zero.
Lemma 7.1
Assume that the hypotheses (7.1), (7.2), and (7.3) are satisfied. Define
and
Then, we have that
Proof
Observe that (7.3) implies that \(\left| \varvec{\mu }^\pm _o \right| \le \tfrac{5}{4} \varvec{\omega }_o\). Furthermore, we have
since \(\varvec{\omega }_o = \varvec{\mu }^+_o - \varvec{\mu }^-_o\). Suppose first that (7.6)\(_1\) holds true. Then, we have \(\tfrac{1}{2} \varvec{\omega }_o \le \varvec{\mu }^+_o \le \tfrac{5}{4} \varvec{\omega }_o\). In this case, we distinguish between the alternatives
When (7.7)\(_1\) holds true, we apply Lemma 5.4. Since \(|\varvec{\mu }^-_o|\le \tfrac{5}{4} \varvec{\omega }_o\), this yields
On the other hand, if (7.8)\(_2\) holds true, we may apply Lemma 6.4 and obtain
Clearly \(\lambda \le \tfrac{1}{2}\) and in the case \(0<q<1\) we can estimate
by \(\varvec{\omega }_1 \ge \delta \varvec{\omega }_o\). If \(q>1\) and \(\varvec{\omega }_1 = \delta \varvec{\omega }_o\) the same inequality holds true. If \(q > 1\) and \(\varvec{\omega }_1 = 2 {{\,\textrm{osc}\,}}_{Q_o} \psi \) holds, we use that \(2 {{\,\textrm{osc}\,}}_{Q_o} \psi \le \varvec{\omega }_o\) by assumption. Hence,
follows in any case. Therefore, we have that either
or
This completes the proof in case (7.6)\(_1\). Now, suppose that (7.6)\(_2\) holds true. Then, we have that \(- \tfrac{5}{4} \varvec{\omega }_o \le \varvec{\mu }^-_o \le - \tfrac{1}{2} \varvec{\omega }_o\) and distinguish between the alternatives
When (7.7) holds true, we may apply Lemma 5.3, since \(|\varvec{\mu }^+_o|\le \tfrac{5}{4}\varvec{\omega }_o\). This implies
On the other hand if (7.8)\(_2\) holds true, we apply Lemma 6.3 to obtain
By using similar estimates as above, we conclude that
which finishes the proof. \(\square \)
Up next, we will prove a similar result in the case where u is bounded away from zero and positive.
Lemma 7.2
Assume that (7.1), (7.2), and (7.4) hold true. For the sequence of cylinders
we define
Then, for any \(i \in \mathbb {N}_0\) there holds
Proof
First, observe that
for any \(i \in \mathbb {N}_0\). Define
for every \(i \in \mathbb {N}\). Now by the assumptions, we already have that \(\varvec{\omega }_1 \le \varvec{\omega }_o\) holds true. By induction it follows directly that
for all \(i \in \mathbb {N}_0\). Since (7.4) is equivalent to \(\varvec{\mu }^+_o < 5 \varvec{\mu }^-_o\), we have that \(\theta ^\frac{1}{q-1} = \varvec{\mu }^+_o < 5 \varvec{\mu }^-_o \le 5 \varvec{\mu }^-_i \le 5 \varvec{\mu }^+_i \). Further, we know that \(\varvec{\mu }^+_i = \varvec{\mu }^-_i + \varvec{\omega }_i \le {{\,\textrm{sup}\,}}_{Q_o} u + \varvec{\omega }_o \le 2 \varvec{\mu }^+_o = 2 \theta ^\frac{1}{q-1}\). Therefore, we find that
for any \(i \in \mathbb {N}_0\). Moreover, we have that
for every \(i \in \mathbb {N}_0\). Assume that \({{\,\textrm{osc}\,}}_{Q_i} u \le \varvec{\omega }_i\) for some \(i \in \mathbb {N}\). For \(i=0\), this clearly holds by (7.1). Then, in particular we have that \(\varvec{\mu }^+_i = \varvec{\mu }^-_i + \varvec{\omega }_i \ge {{\,\textrm{inf}\,}}_{Q_i} u + {{\,\textrm{osc}\,}}_{Q_i} u = {{\,\textrm{sup}\,}}_{Q_i} u\). Now, we distinguish between the alternatives
When the first alternative holds true, by Lemma 5.2 we obtain that
On the other hand if the second alternative holds true, Lemma 6.4 implies that
for some \(a = a(n,q,C_o,C_1) \in (0,\tfrac{1}{64}]\). Recalling that \(\delta = 1-a\), we see that in both cases
which completes the proof. \(\square \)
Finally we state and prove a similar lemma in the case where u is bounded away from zero and negative. Observe that the problem is not symmetric in the case where the solution u is above and the case where it is below zero, since the obstacle is restricting the behavior of u only from below. Thus, in Lemma 7.3 we cannot proceed analogously to Lemma 7.2 by defining \(\varvec{\mu }^+_i = \sup _{Q_i} u\) and \(\varvec{\mu }^-_i = \varvec{\mu }^+_i - \varvec{\omega }_i\), since the condition \(\sup _{Q_{\rho _i,\theta \rho _i}} \psi \le \tfrac{1}{2} \left( \varvec{\mu }^+_i + \varvec{\mu }^-_i \right) \) needed for the application of Lemma 5.1 could be violated. Hence, we use a different approach.
Lemma 7.3
Assume that the hypotheses (7.1), (7.2), and (7.5) hold. For the sequence of cylinders
we define
Then, for any \(i \in \mathbb {N}_0\) there holds
Proof
First, observe that
for any \(i \in \mathbb {N}_0\). Define
for every \(i \in \mathbb {N}\). Now by the assumptions, we already have that \(\varvec{\omega }_1 \le \varvec{\omega }_o\) holds true and by induction it directly follows that
for all \(i \in \mathbb {N}_0\), where we have used that \(\{\varvec{\mu }^+_i\}_{i\in \mathbb {N}_0}\) is a nonincreasing sequence by definition. Since (7.5) is equivalent to \(\varvec{\mu }^-_o > 5 \varvec{\mu }^+_o\) and \(\{\varvec{\mu }^+_i\}_{i\in \mathbb {N}_0}\) is nonincreasing, we have that \(\theta ^\frac{1}{q-1} = - \varvec{\mu }^-_o < -5 \varvec{\mu }^+_o \le -5 \varvec{\mu }^-_i \) and that \(\theta ^\frac{1}{q-1} = -\varvec{\mu }^-_o \ge - \varvec{\mu }^-_i\); i.e., we find that
for any \(i \in \mathbb {N}_0\). Up next, we show that the condition \(\sup _{Q_i}\psi \le \tfrac{1}{2} \left( \varvec{\mu }^+_i + \varvec{\mu }^-_i \right) \) holds true for all \(i\in \mathbb {N}_0\). For \(i=0\) this is part of hypothesis (7.2). Suppose that this holds for some \(i \in \mathbb {N}\). On the one hand if \(\varvec{\mu }^+_{i+1} = \varvec{\mu }^-_{i+1} + \varvec{\omega }_{i+1}\), we have that
On the other hand if \(\varvec{\mu }^+_{i+1} = \varvec{\mu }_{i}^+\), by the induction assumption and the property that \(\varvec{\mu }^-_i \le \varvec{\mu }^-_{i+1}\) we may estimate
Next, we want to show that for every \(i \in \mathbb {N}\), there holds
For \(i =1\), (7.9)\(_1\) clearly holds by assumption. Now, we consider the alternatives
If the first alternative holds true, we may apply Lemma 5.1. On the other hand, if the second alternative holds true, we use Lemma 6.3. In both cases, we find that
where the last inequality holds by definition of \(\varvec{\omega }_1\), and \(\delta = 1-a\) with the constant a from Lemma 6.3. This takes care of the case \(i=1\). Now let us suppose that (7.9) holds for some \(i\in \mathbb {N}\). It follows that either we have
with assumption (7.9)\(_2\) or that
with assumption (7.9)\(_1\). The two inequalities above already prove (7.9)\(_1\). Let us define \( \tilde{\varvec{\omega }}_i = \varvec{\mu }^+_i - \varvec{\mu }^-_i \le \varvec{\omega }_i\). Now, we will use the alternatives
In the first case, we apply Lemma 5.1 with \( \tilde{\varvec{\omega }}_i\) in place of \(\varvec{\omega }_i\). This implies that
Since \(Q_{i+1} \subset Q_{\frac{\rho _i}{2}, \theta \left( \frac{\rho _i}{2} \right) ^2} \subset Q_i\), we have
since \(\tilde{\varvec{\omega }}_i \le \varvec{\omega }_i\) and by definition of \(\varvec{\omega }_{i+1}\). If (7.10)\(_2\) holds true, we use Lemma 6.3, which gives us
for some \(a = a(n,q,C_o,C_1) \in (0,\tfrac{1}{64}]\). Recalling that \(\delta = 1-a\), similarly to (7.11) we obtain
Now (7.11) and (7.12) imply (7.9)\(_2\), which completes the proof. \(\square \)
8 Proof of Theorem 1.1
In the following, we assume that u is globally bounded for ease of notation. However, the argument holds for a locally bounded weak solution u by restricting to a compact subset of \(\Omega _T\). Thus, we can assume that
by using the rescaling argument from Lemma A.1 with \(M = 2 \Vert u\Vert _\infty \).
Assume that \(\psi \in C^{0; \beta , \frac{\beta }{2}}(\Omega _T)\) for the exponent \(\beta \in (0,1)\), i.e.,
Let \(\epsilon = \frac{2 \beta (1-q)_+}{2 + \beta (1-q)_+} \in [0,\tfrac{2}{3})\) and \(\gamma _o = \frac{2 \beta }{2 + \beta (1-q)_+} = \big ( 1 - \frac{\epsilon }{2} \big ) \beta \in (0, \beta ]\). Observe that \(\varepsilon = 0\) and \(\gamma _o = \beta \) in the singular case \(q>1\). Further, consider an arbitrary point \(z_o = (x_o, t_o) \in \Omega _T\) and let \(R \in (0,1)\) be so small that \(Q_{R, R^{2-\epsilon }}(z_o) \Subset \Omega _T\). In the following, we omit \(z_o\) from our notation for simplicity. Next, we consider the function
which is continuous and increasing. By the assumption \(\psi \in C^{0; \beta , \frac{\beta }{2}}(\Omega _T)\), the choice of \(\epsilon \) and the fact that \(\rho \in (0,1)\) we obtain that
Thus, we conclude that u is Hölder continuous at \((x_o, t_o)\) in the case that the bound
holds. In order to prove the Hölder continuity of u in the opposite case, observe that if the preceding inequality is false, then there exists \(\rho _o \in (0,R]\) such that
since the map \(\rho \mapsto {{\,\textrm{osc}\,}}_{Q_{\rho , \rho ^{2-\epsilon }}} u\) is increasing, the map \(\rho \mapsto \Psi (\rho )\) is continuous and increasing and \({{\,\textrm{osc}\,}}_{Q_{R,R^{2-\epsilon }}} u \le 1 \le R^{-\gamma _o} \Psi (R)\). For this choice of \(\rho _o\), we define
In the case \(0<q<1\) we define \(\theta _o := \varvec{\omega }_o^{q-1}\). Further, we compute that
by definition of \(\rho _o\). In the singular case \(q>1\), we define
By taking into account (8.1), we conclude that in any case \(\theta _o \le 1\) when \(q > 1\). Hence, we have the set inclusion
Since \(u \ge \psi \) a.e. in \(\Omega _T\) and by the choice of \(\rho _o\), we deduce that
By the definition of \(\Psi \) and (8.2)\(_1\) we infer
In the following, the strategy is to construct a sequence of shrinking cylinders \(\{Q_i\}_{i \in \mathbb {N}_0}\) with common vertex \(z_o\), such that the oscillation of u in these cylinders can be reduced in a quantitative way when passing from \(Q_{i}\) to the subsequent cylinder \(Q_{i+1}\). Exactly one of the three possible cases will hold in each cylinder of the sequence: u is near zero, u is above and away from zero or u is below and away from zero. If u is near zero in \(Q_i\), any of the three cases may hold in \(Q_{i+1}\). However, if either of the cases where u is bounded away from zero holds in \(Q_i\), the same case will also hold in \(Q_{i+1}\) and in every subsequent cylinder. Up next, \(\varvec{\mu }^+_i\) will roughly denote an upper bound for u, \(\varvec{\mu }^-_i\) a lower bound for u and \(\varvec{\omega }_i\) an upper bound for the oscillation of u in the cylinder \(Q_i\). Precise definitions are given in the following subsections. Now, we proceed as follows: Suppose that \(i_o \in \mathbb {N}_0 \cup \{\infty \}\) is the largest index for which we have
for all \(i \in \{ 0,1, \ldots ,i_o -1 \}\). This means that up to the index \(i_o - 1\), we apply the reduction in oscillation for the case where u is near zero (see Sect. 8.1). For every \(i \ge i_o\), it then follows that either
holds true and we use the results on a reduction in oscillation for either one of the cases where u is away from zero (see Sects. 8.2–8.3). Observe that it is also possible that \(i_o = 0\), which means that the case where u is near zero never occurs, or that \(i_o = \infty \) and the iteration is carried out for u near zero completely.
8.1 Reduction in oscillation near zero
Suppose that \(i_o > 0\), otherwise we can skip this part and move directly to either Sect. 8.2 or 8.3 depending on which case in (8.6) holds true. For \(i \in \{ 1,2,\ldots ,i_o \}\), define
This implies
and
for every \(i \in \{1,2,\ldots ,i_o\}\). Now, we claim that
Clearly, this holds true by definitions when \(i=0\). Suppose that the statement holds true for some \(i < i_o\). Then, we have that
by assumption. Now, we are in a point of using Lemma 7.1, which implies that
which proves (8.7).
8.2 Reduction in oscillation above and away from zero
Suppose that \(i_o \in \mathbb {N}_0\) is the first index for which there holds that
Now, we define \(\theta _* = \left( \varvec{\mu }^+_{i_o} \right) ^{q-1}\). In the case \(0<q<1\) it directly follows that \(\theta _* \le \left( \varvec{\omega }_{i_o} \right) ^{q-1} = \theta _{i_o}\). If \(q > 1\) and \(i_o = 0\) we have \(\theta _* = \theta _o\). If \(i_o \ge 1\) we deduce that
by using (8.5) for the index \(i_o-1\). Since (8.8) is equivalent to \(\varvec{\mu }^+_{i_o} < 5 \varvec{\mu }^-_{i_o}\), this implies that
Now, we can deduce the bound \(\theta _* < \left( \tfrac{25}{4 \delta } \varvec{\omega }_{i_o} \right) ^{q-1} = \left( \tfrac{25}{4 \delta } \right) ^{q-1} \theta _{i_o}\) when \(q > 1\), which is taken into account in the following construction. For the cylinders \(i > i_o\) let
and let \(Q^*_{i_o}= Q_{{\hat{\rho }}_{i_o},\theta _* {\hat{\rho }}_{i_o}^2} \subset Q_{i_o}\), where \(Q_{i_o}\) is the cylinder obtained in the last section after application of the last iteration step or it is \(Q_{i_o} = Q_o\) if \(i_o = 0\). Now, we clearly have that \(Q_{i_o} \supset Q_{i_o}^{*} \supset Q_{i_o+1} \supset \ldots \) and
Further, we find that
where the last inequality follows from Sect. 8.1 if \(i_o > 0\) and from (8.3) if \(i_o = 0\). Finally, we obtain that
which follows from Sect. 8.1 if \(i_o>0\), and from (8.4) if \(i_o = 0\). Now, we are in the position to use Lemma 7.2, which implies
8.3 Reduction in oscillation below and away from zero
Suppose that \(i_o \in \mathbb {N}_0\) is the first index for which there holds that
Now, we define \(\theta _* = \left( - \varvec{\mu }^-_{i_o} \right) ^{q-1}\). If \(0<q<1\), it follows that \(\theta _* \le \left( \varvec{\omega }_{i_o} \right) ^{q-1} = \theta _{i_o}\). In the case \(q> 1\), \(\theta _* = \theta _{i_o}\) if \(i_o = 0\), and if \(i_o > 0\) we deduce that
by using the condition (8.5) for the index \(i_o-1\) in the penultimate inequality. Since (8.9) is equivalent to \(5 \varvec{\mu }^+_{i_o} < \varvec{\mu }^-_{i_o}\), this implies
For the cylinders \(i > i_o\) we define
and let \(Q^*_{i_o}= Q_{{\hat{\rho }}_{i_o},\theta _* {\hat{\rho }}_{i_o}^2} \subset Q_{i_o}\), where \(Q_{i_o}\) is the cylinder obtained in Sect. 8.1 after the last iteration step. Analogously to the Sect. 8.2, we are now in the position to use Lemma 7.3, which implies
8.4 Proof of the oscillation decay estimate
We define
where \({\hat{\rho }}_i := \rho _i\) for \(i < i_o\). We claim that
for any \(i \in \mathbb {N}_0\).
Recall that either \(\theta _* = \left( \varvec{\mu }^+_{i_o} \right) ^{q-1}\) if (8.8) holds, or \(\theta _* = \left| \varvec{\mu }^-_{i_o} \right| ^{q-1}\) if (8.9) holds. Indeed, \(r_i \le {\hat{\rho }}_i\) and if \(q>1\) we know that \(\theta _i = \varvec{\omega }_i^{q-1} \ge \left( \delta ^i \varvec{\omega }_o \right) ^{q-1}\) for \(i \le i_o\) and \(\theta _* \ge \left( \tfrac{1}{4} \varvec{\omega }_{i_o} \right) ^{q-1} \ge \left( \tfrac{\delta ^{i_o}}{4} \varvec{\omega }_o \right) ^{q-1}\) if \(i_o > 0\). If \(q > 1\) and \(i_o = 0\), we have \(\theta _* > \left( \tfrac{1}{4} \varvec{\omega }_o \right) ^{q-1}\). Moreover, if \(0<q<1\) we have that \(\theta _i = \varvec{\omega }_i^{q-1} \ge \varvec{\omega }_o^{q-1}\) for \(i \le i_o\) and \(\theta _* \ge \left( \tfrac{25}{4} \varvec{\omega }_{i_o-1} \right) ^{q-1} \ge \left( \tfrac{25}{4} \varvec{\omega }_o \right) ^{q-1}\) by definition of \(i_o\) if \(i_o > 0\). In case \(0<q<1\) and \(i_o = 0\), we can use (8.1) such that \(\theta _* \ge 1\). Therefore, we find that
When \(i-1-j\le i_o\), by the fact that \(\rho _{i-1-j} = \lambda ^{i-1-j} \rho _o\) and the definition of \(\lambda \) we estimate
Then let \(i-1-j > i_o\). Recall that by construction we have \(\theta _* \le \theta _{i_o} \le \left( \delta ^{i_o} \varvec{\omega }_o\right) ^{q-1}\) if \(0< q < 1\), and in case \(q > 1\) we can simply use the rescaling to estimate \(\theta _*\le 1\). By the definitions of \(\lambda \) and \({\hat{\lambda }}\) we obtain in a similar way as above that
Using the estimates above in (8.10) gives us
Setting
we conclude from the preceding inequality that
By the fact that, \(i \tau ^{i-1} \le c(\tau ) \sqrt{\tau }^i\), we infer
for a constant \(c = c(n,q,C_o,C_1,\beta )\). Let us define \(\eta = \left( \tfrac{\delta }{5} \right) ^{(q-1)_+} \lambda \) and
depending on n, q, \(C_o\), \(C_1\) and \(\beta \) and observe that \(\gamma _1 \le \frac{\log \tau }{2 \log \eta } \le \frac{\log \delta }{2 \log \eta }\). Now, we have
Observe that (8.2) implies
Let \(q > 1\). Recall that in this case \(\gamma _o = \beta \) and by using \(\varvec{\omega }_o \le 1\) we obtain
Furthermore, we have that
for \(i \in \mathbb {N}\) by definitions of \(r_i\), \({\hat{\rho }}_i\) and \(\eta \). Now, since \(\varrho _o \in (0,1)\) and \(\beta - \gamma _1 - \gamma _1\beta \frac{q-1}{2} \ge 0\) by definition of \(\gamma _1\), by using the estimate above and (8.11)\(_1\) we have
for \(c = c(n,q,C_o,C_1,\beta )\).
Then let \(0<q<1\). Observe that by using the definition of \(r_i\) and \(\varvec{\omega }_o \in (0,1]\) we have \(\eta ^i \le \frac{{\hat{\rho }}_i}{\rho _o} \le c(q) \frac{r_i}{\rho _o}\). By using this fact together with \(\varvec{\omega }_o^{q-1} \le \rho _o^{-\varepsilon }\), (8.11)\(_2\), \(\varrho _o \in (0,1)\) and \(\gamma _1 \le \gamma _o\) we can conclude
for a constant \(c = c(n,q,C_o,C_1,\beta )\). Thus, for all \(q > 0\) we have
We conclude this section by showing that the last estimate holds for an arbitrary radius \(r \in (0,R]\). First, let us consider \(r \in (0,r_o)\). Choose \(i\in \mathbb {N}_0\) such that \(r_{i+1}< r < r_i\). By (8.12) and the fact that \(\varvec{\omega }_o \le 1\), we find that
for a constant \(c = c(n,q,C_o,C_1,\beta )\). Next, let us assume that \(r \in [r_o, \rho _o)\). By (8.11)\(_2\) and since \(\rho _o \le \big ( \frac{25}{4} \big )^\frac{1-q}{2} r_o\) when \(0<q<1\) and \(r_o \ge c(q) \rho _o^{1+ \beta \frac{q-1}{2}}\) implying \(r_o^{\gamma _1} \ge c \rho _o^{\gamma _o}\) by \(\gamma _o \ge \gamma _1 + \gamma _1\beta \frac{q-1}{2}\) when \(q > 1\), we obtain that
In the remaining case \(r\in [\rho _o, R]\), we have that
Altogether, recalling that we have omitted \((x_o,t_o)\) in our notation, we infer
8.5 Quantitative Hölder estimate
Denote the parabolic boundary of \(\Omega _T\) by \(\Gamma \) and consider \((x_1,t_1), (x_2,t_2) \in K\) for a compact subset \(K \Subset \Omega _T\). Without loss of generality, assume that \(t_2 > t_1\). Define the (non-intrinsic) parabolic distance
Since
we can prove quantitative Hölder estimates with respect to the space and time variables separately. We only give the proof of the latter, since the proof of the former is analogous. To this end, note that for
we have that
We remark that in the obstacle-free case, starting with any intrinsic cylinder \(Q_{R, \varvec{\omega }_o^{q-1} R^2} \Subset \Omega _T\) arguments as in Sects. 8.1 – 8.4 can be repeated and thus an analogous oscillation decay estimate to (8.13) is derived without assuming that \(Q_{R, \varvec{\omega }_o^{q-1} R^2}\) is contained in any specific non-intrinsic cylinder. In contrast, in the present situation the construction of intrinsic cylinders contained in a non-intrinsic cylinder of the form \(Q_{R,R^{2-\epsilon }}\) in the beginning of Sect. 8 seems to be unavoidable in the degenerate case \(0<q<1\), since it is used to deal with the factor \(\varvec{\omega }_o^\frac{\beta (q-1)}{2}\), which is related to the oscillation of the obstacle \(\psi \) in intrinsic cylinders of the form \(Q_{r, \varvec{\omega }_o^{q-1} r^2}\).
Now, we distinguish between the cases
If (8.14)\(_1\) holds, choose \(r \in (0,R)\) such that \(t_2 - t_1 = r^2\). Applying the oscillation decay estimate (8.13) in \(Q_{r,r^2}(x_2,t_2)\) and recalling the definition of \(\gamma _o\) leads to
In the case (8.14)\(_2\), we use (8.1) and the facts that \(R\le 1\) and \(\gamma _1 \le \gamma _o\) to estimate
Together with analogous estimates for the space variables, we arrive at
for bounded local weak solutions u of the obstacle problem satisfying (8.1). Taking the rescaling argument from Lemma A.1 with \(M=2 \Vert u\Vert _\infty \) into account, we infer the quantitative Hölder estimate (1.3) with \(\gamma =\gamma _1\) for general bounded local weak solutions u in the sense of Definition 2.1. In particular, note that rescaling does not affect the Hölder exponent \(\beta \) of the rescaled obstacle function \({\widetilde{\psi }}\) and there holds \([{\widetilde{\psi }}]_{C^{0;\beta ,\frac{\beta }{2}}} = \frac{1}{M} [\psi ]_{C^{0;\beta ,\frac{\beta }{2}};q}\), see Appendix A.
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K. Moring has been supported by the Magnus Ehrnrooth Foundation and Foundation for Aalto University Science and Technology.
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Appendix A: Rescaling argument
Appendix A: Rescaling argument
Let \(M>0\) and consider the rescaled functions
together with
for \((x,t) \in \Omega _{\widetilde{T}} := \Omega \times (0,\widetilde{T}) := \Omega \times (0, M^{1-q} T)\), such that the vector field \(\widetilde{{\textbf{A}}}\) satisfies the same structure conditions as \({\textbf{A}}\).
Lemma A.1
Let \({\tilde{u}}\) and \({\widetilde{\psi }}\) be defined as in (A.1) and \(\widetilde{{\textbf{A}}}\) as in (A.2). Then, \({\tilde{u}}\) is a weak solution to the obstacle problem with obstacle \({\widetilde{\psi }}\) and
in the sense of Definition 2.1.
Proof
For \({\widetilde{\psi }} \in C^0(\Omega _{\widetilde{T}})\), we compute that
Hence, we have that \({\widetilde{\psi }} \in C^{0;\beta ,\frac{\beta }{2}}(\Omega _{\widetilde{T}})\) and that \([{\widetilde{\psi }}]_{C^{0;\beta ,\frac{\beta }{2}}} = \frac{1}{M} [\psi ]_{C^{0;\beta ,\frac{\beta }{2}};q}\). Further, it is clear that there holds \({\tilde{u}} \ge {\widetilde{\psi }}\) a.e. in \(\Omega _{\widetilde{T}}\) and we compute that
i.e., we find that \({\tilde{u}} \in K_{{\widetilde{\psi }}}(\Omega _{\widetilde{T}})\). Now, we consider \({\widetilde{\varphi }} \in C^\infty _0(\Omega _{\widetilde{T}};\mathbb {R}_{\ge 0})\) and \({\tilde{v}} \in K_{{\widetilde{\psi }}}'(\Omega _{\widetilde{T}})\). First, observe that \(\varphi (x,t) := {\widetilde{\varphi }}(x, M^{1-q} t) \in C^\infty _0(\Omega _T;\mathbb {R}_{\ge 0})\). Furthermore, we state that \(v(x,t) := M {\tilde{v}}(x, M^{1-q} t)\) is an admissible comparison map related to u and \(\psi \). To this end, check that \(v \in C^0((0,T); L^{q+1}_{{{\,\textrm{loc}\,}}}(\Omega )) \cap L^2_{{{\,\textrm{loc}\,}}}(0,T; H^1_{{{\,\textrm{loc}\,}}}(\Omega ))\) and
for a.e. \((x,t) \in \Omega _T\). Moreover, compute that
Altogether, this implies that \(v \in K_\psi '(\Omega _T)\). With these considerations at hand, a straightforward computation shows that
since u is a weak solution associated with the obstacle \(\psi \) in the sense of Definition 2.1.\(\square \)
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Moring, K., Schätzler, L. On the Hölder regularity for obstacle problems to porous medium type equations. J. Evol. Equ. 22, 81 (2022). https://doi.org/10.1007/s00028-022-00840-4
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DOI: https://doi.org/10.1007/s00028-022-00840-4