Abstract
In this paper, we study two related features of the regularity of the weak solutions to the following strongly degenerate parabolic equation
where \(\Omega \) is a bounded domain in \(\mathbb {R}^{n}\) for \(n\ge 2\), \(p\ge \text {and}\, T>0\). We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that \(f\in L^{2}_{\textrm{loc}}\left( \Omega _T\right) \). This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f.
Similar content being viewed by others
1 Introduction
In this paper, we study the regularity properties of weak solutions \(u:\Omega _{T}\rightarrow \mathbb {R}\) to the following parabolic equation
which appears in gas filtration problems taking into account the initial pressure gradient. For a precise description of this motivation we refer to [1] and [3, Sect. 1.1].
The main feature of this equation is that it possesses a wide degeneracy, coming from the fact that its modulus of ellipticity vanishes at all points where \(\left| Du\right| \le 1\) and hence its principal part behaves like a p-Laplacian operator only for large values of \(\left| Du\right| \).
In this paper we address two interrelated aspects of the regularity theory for solutions to parabolic problems, namely the higher differentiability and the higher integrability of the gradient of the weak solutions to (1.1), with the main aim of weakening the assumption on the datum f with respect to the available literature.
These questions have been exploited in case of non-degenerate parabolic problems with quadratic growth by Campanato in [12], by Duzaar et al. in [19] in case of superquadratic growth, while Scheven in [24] faced the subquadratic growth case. In the above mentioned papers, the problem has been faced or in case of homogeneous equations or considering sufficiently regular datum. It is worth mentioning that the higher integrability of the gradient of the solution is achieved through an interpolation argument, once its higher differentiability is established.
This strategy has revealed to be successful also for widely degenerate equations as in (1.1). Indeed, the higher integrability of the spatial gradient of weak solutions to equation (1.1) has been proven in [3], under suitable assumptions on the datum f in the scale of Sobolev spaces.
We’d like to recall that a common feature for nonlinear problems with growth rate \(p>2\) is that the higher differentiability is proven for a nonlinear expression of the gradient which takes into account the growth of the principal part of the equation.
Indeed, already for the non-degenerate p-Laplace equation, the higher differentiability refers to the function \(V_{{\mu },p}\left( Du\right) =\left( {\mu ^2}+\left| Du\right| ^2\right) ^\frac{p-2}{4}Du\). In case of widely degenerate problems, this phenomenon persists, and higher differentiability results, both for the elliptic and the parabolic problems, hold true for the function \(H_\frac{p}{2}\left( Du\right) =\left( \left| Du\right| -1\right) _+^{\frac{p}{2}}\displaystyle \frac{Du}{\left| Du\right| }\).
It is worth noticing that, as it can be expected, this function of the gradient doesn’t give information on the second order regularity of the solutions in the set where the equation degenerates. Actually, since every 1-Lipschitz continuous function is a solution to the elliptic equation
where \(H_{p-1}\left( Du\right) =\left( \left| Du\right| -1\right) _+^{p-1}\displaystyle \frac{Du}{\left| Du\right| }\), no more than Lipschitz regularity can be expected.
Moreover, it is well known that (already for the degenerate p-Laplace equation, with \(p>2\)) a Sobolev regularity is required for the datum f in order to get the higher differentiability of the solutions, when such higher differentiability comes from the existence of the weak derivatives of the function \(V_{0,p}(Du)\equiv V_{p}(Du)=\left| Du\right| ^{\frac{p-2}{2}}Du\) (see, for example, [11] for elliptic and [3] for parabolic equations). Actually, the sharp assumption for the datum in the elliptic setting has been determined in [11] as a fractional Sobolev regularity suitably related to the growth exponent p and the dimension n.
On the other hand, in [13, 14], the authors proved that the assumption \(f\in L^2\) is sufficient both in the elliptic and in parabolic setting to prove the \(W^{1,2}\)-regularity of \(\left| Du\right| ^{p-2}Du\), which is a different function of the gradient with respect to \(V_p\left( Du\right) \). Note, however, that \(\left| D\left( V_p\left( Du\right) \right) \right| \approx \left| Du\right| ^{\frac{p-2}{2}}\left| D^2u\right| \) while \(\left| D\left( \left| Du\right| ^{p-2}Du\right) \right| \approx \left| Du\right| ^{p-2}\left| D^2u\right| \) and therefore, in case \(p\ge 2\), if \(\left| Du\right| \) approaches zero, i.e. when we are closed to the degeneracy set, the \(W^{1,2}\)-regularity of \(V_p\left( Du\right) \) implies that of \(\left| Du\right| ^{p-2}Du\).
The main aim of this paper is to show that, without assuming any kind of Sobolev regularity for the datum, but assuming only \(f\in L^2\), we are still able to obtain the differentiability of a function of the weak solutions that behaves as \(V_p\left( Du\right) \) outside a set larger than the degeneracy set of the problem. We would like to emphasize that, while for the p-Laplace equation the degeneracy appears for \(p>2\), here, even in case \(p=2\), under a \(L^2\) integrability assumption on the datum f, the local \(W^{2,2}\) regularity of the solutions cannot be obtained.
Actually, our main result is the following.
Theorem 1.1
Let \(n\ge 2\), \(p\ge 2\) and \(f\in L^{2}_{\textrm{loc}}\left( \Omega _T\right) \). Moreover, let us assume that
is a weak solution to (1.1). Then, for any \(\delta \in (0,1)\), we have
where
Moreover the following estimate
holds for any \(R>0\) such that \(Q_R=Q_R\left( z_{0}\right) \Subset \Omega _T\).
As already mentioned, the weak solutions of (1.1) are not twice differentiable, and hence it is not possible in general to differentiate the equation to estimate the second derivative of the solutions. We overcome this difficulty by introducing a suitable family of approximating problems whose solutions are regular enough by the standard theory [17]. The major effort in the proof of Theorem 1.1 is to establish suitable estimates for the solutions of the regularized problems that are uniform with respect to the approximation’s parameter. Next, we take advantage from these uniform estimates in the use of a comparison argument aimed to bound the difference quotient of \(\mathcal {G}_\delta \left( \left( \left| Du\right| -\delta -1\right) _+\right) \).
The main difference with respect to the arguments used in [3] is in the derivation of the a priori estimates for the terms involving the datum f. Indeed, if f has a Sobolev regularity, the terms of the estimate coming from the inhomogeneity of the equation can be controlled using the informations on the integrability of the gradient of the solution. Here, the terms coming from the inhomogeneity behave as the second derivatives of the solution and their \(L^2\) integrability is known to fail already for solutions to less degenerate equations. Hence here, differently from [3], we need to construct a suitable function of the gradient with weak derivatives in \(L^2\) that, on the one hand, vanishes on a ball larger than \(B_1\) and on the other is comparable with the ellipticity bounds of our equation.
Hence, roughly speaking, due to the weakness of our assumption on the datum, we only get the higher differentiability of a nonlinear function of the gradient of the solutions that vanishes in a set which is larger with respect to that of the degeneracy of the problem. This is quite predictable, since the same kind of phenomenon occurs in the setting of widely degenerate elliptic problems (see, for example [15]).
Anyway, as a consequence of the higher differentiability result in Theorem 1.1, since the gradient stays bounded in the degeneracy set, we establish a higher integrability result for the spatial gradient of the solution to Eq. (1.1), which is the following.
Theorem 1.2
Under the assumptions of Theorem 1.1, we have
with the following estimate
for every parabolic cylinder \(Q_{{2\rho }}\left( z_{0}\right) \Subset \Omega _T\), with a constant \(c=c(n,p).\)
The proof of previous Theorem consists in using an interpolation argument with the aim of establishing an estimate for the \(L^{p+\frac{4}{n}}\) norm of the gradient of the solutions to the approximating problems that is preserved in the passage to the limit.
We would like to mention that the elliptic version of our equation naturally arises in optimal transport problems with congestion effects, and the regularity properties of its weak solutions have been widely investigated. More precisely, Lipschitz regularity results can be found in [4, 5, 9, 16, 20], while we refer to [2, 11] for higher differentiability and to [6] for a continuity result for the gradient of weak solutions.
We conclude stressing that, for sake of clarity, we confine ourselves to Eq. (1.1), but we believe that our techniques apply as well to a general class of equations with an analogous widely degenerate structure and also to solutions of systems of widely degenerate parabolic equations.
2 Notations and preliminaries
In this paper we shall denote by C or c a general positive constant that may vary on different occasions. Relevant dependencies on parameters will be properly stressed using parentheses or subscripts. The norm we use on \(\mathbb {R}^{n}\) will be the standard Euclidean one and it will be denoted by \(\left| \,\cdot \,\right| \). In particular, for the vectors \(\xi ,\eta \in \mathbb {R}^{n}\), we write \(\langle \xi ,\eta \rangle \) for the usual inner product and \(\left| \xi \right| :=\langle \xi ,\xi \rangle ^{\frac{1}{2}}\) for the corresponding Euclidean norm.
For points in space-time, we will use abbreviations like \(z=(x,t)\) or \(z_{0}=\left( x_{0},t_{0}\right) \), for spatial variables x, \(x_{0}\in \mathbb {R}^{n}\) and times t, \(t_{0}\in \mathbb {R}\). We also denote by \(B\left( x_{0},\rho \right) =B_{\rho }\left( x_{0}\right) =\left\{ {x\in \mathbb {R}^{n}:\left| x-x_{0}\right| <\rho }\right\} \) the open ball with radius \(\rho >0\) and center \(x_{0}\in \mathbb {R}^{n}\); when not important, or clear from the context, we shall omit to indicate the center, as follows: \(B_{\rho }\equiv B\left( x_{0},\rho \right) \). Unless otherwise stated, different balls in the same context will have the same center. Moreover, we use the notation
for the backward parabolic cylinder with vertex \(\left( x_{0},t_{0}\right) \) and width \(\rho \). We shall sometimes omit the dependence on the vertex when the cylinders occurring share the same vertex. Finally, for a cylinder \(Q=A\times \left( t_{1},t_{2}\right) \), where \(A\subset \mathbb {R}^{n}\) and \(t_{1}<t_{2}\), we denote by
the usual parabolic boundary of Q, which is nothing but its standard topological boundary without the upper cap \(A\times \left\{ {t_{2}}\right\} \).
We now recall some tools that will be useful to prove our results. For the auxiliary function \(H_{\lambda }:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) defined as
where \(\lambda >0\) is a parameter, we record the following estimates (see [10, Lemma 4.1]).
Lemma 2.1
If \(2\le p<\infty \), then for every \(\xi ,\eta \in \mathbb {R}^{n}\) it holds
We also recall the following
Lemma 2.2
Let \(\xi ,\eta \in \mathbb {R}^{k}\) with \(|\xi |>1\). Then, we have
and
For the proof see [6, Lemma 2.8].
Previous Lemma implies that the operator \(D_\xi H_{p-1}(\xi )\) is controlled with two different functions of \(\left( \left| \xi \right| -1\right) _+,\) and so our problem can be also viewed as a very particular case of parabolic problems with non-standard growth, whose study started with the papers [7, 8].
Definition 2.3
With the use of (2.1), a function \(u\in C^{0}\left( 0,T;L^{2}\left( \Omega \right) \right) \cap L^{p}\left( 0,T;W^{1,p}\left( \Omega \right) \right) \) is a weak solution of Eq. (1.1) if
for every \(\varphi \in C_{0}^{\infty }(\Omega _{T})\).
We shall also use the well known auxiliary function \(V_{1, p}:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) defined as
where \(p\ge 2\), for which we have the following
Lemma 2.4
For every \(\xi ,\eta \in \mathbb {R}^{n}\) there hold
We refer to [23, Chapter 12] or to [22, Lemma 9.2] for a proof of these fundamental inequalities.
For further needs, we also record the following interpolation inequality whose proof can be found in [18, Proposition 3.1].
Lemma 2.5
Assume that the function \(v:Q_{r}(z_{0})\cup \partial _{\textrm{par}}Q_{r}(z_{0})\rightarrow \mathbb {R}\) satisfies
for some exponents \(1\le p\), \(q<\infty \,\). Then the following estimate
holds true for a positive constant c depending at most on n, p and q.
2.1 Difference quotients
We recall here the definition and some well known properties of the difference quotients (see, for example, [22, Chapter 8]).
Definition 2.6
For every function \(F:\mathbb {R}^{n}\rightarrow \mathbb {R}^N\) the finite difference operator in the direction \(x_{s}\) is defined by
where \(h\in \mathbb {R}\), \(e_{s}\) is the unit vector in the direction \(x_{s}\) and \(s\in \left\{ 1,\ldots ,n\right\} \).
The difference quotient of F with respect to \(x_{s}\) is defined for \(h\in \mathbb {R}\setminus \left\{ 0\right\} \) as
We shall omit the index s when it is not necessary, and simply write \( \tau _{h}F(x)=F(x+h)-F(x) \) and \( \left| \Delta _{h}F(x)\right| =\displaystyle \frac{\left| \tau _{h}F(x)\right| }{|h|} \) for \(h\in \mathbb {R}^n\).
Proposition 2.7
Let \(F\in W^{1,p}\left( \Omega \right) \), with \(p\ge 1\), and let us set
Then:
\(\mathrm {(}a\mathrm {)}\) \(\Delta _{h}F\in W^{1,p}\left( \Omega _{\left| h\right| }\right) \) and
\(\mathrm {(}b\mathrm {)}\) If at least one of the functions F or G has support contained in \(\Omega _{\left| h\right| }\), then
\(\mathrm {(}c\mathrm {)}\) We have
The next result about the finite difference operator is a kind of integral version of Lagrange Theorem (see [22, Lemma 8.1]).
Lemma 2.8
If \(0<\rho <R\), \(\left| h\right| <\displaystyle \frac{R-\rho }{2}\), \(1<p<+\infty \), and \(F\in W^{1, p}\left( B_{R},\mathbb {R}^{N}\right) \), then
Moreover
We conclude this section with the following fundamental result, whose proof can be found in [22, Lemma 8.2]:
Lemma 2.9
Let \(F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{N}\), \(F\in L^{p}\left( B_{R},\mathbb {R}^{N}\right) \) with \(1<p<+\infty \). Suppose that there exist \(\rho \in (0,R)\) and a constant \(M>0\) such that
for every h, with \(\left| h\right| <\displaystyle \frac{R-\rho }{2}\). Then \(F\in W^{1,p}\left( B_{\rho },\mathbb {R}^{N}\right) \) and
Moreover
for each \(s\in \left\{ 1,\ldots ,n\right\} \).
2.2 Some auxiliary functions and related algebraic inequalities
In this section we introduce some auxiliary functions and we list some of their properties, that will be used in what follows.
For any \(k>1\) and for \(s\in [0,+\infty )\), let us consider the function
for which we record the following
Lemma 2.10
Let \(k>1\), and let \(g_k\) be the function defined by (2.3). Then for every \(A,B\ge 0\) the following Young’s type inequality
holds for every parameter \(\alpha \ge 0\) with a constant \(c_\alpha >0\). Moreover, there exists a constant \(c_k>0,\) depending on k, such that
Proof
Since
both the conclusions trivially hold for \(s\le \sqrt{k}.\) Now assume that \(s>\sqrt{k}\) and note that Young’s inequality implies
where we used the explicit expression of \(g'_k(s)\) at (2.6) and the Definition (2.3).
Setting \(h(s)=\displaystyle \frac{s}{k+(s-k)_+^2}\), we can easily check that
and so
Inserting this in (2.7), we get
which, since \(\displaystyle \frac{1}{k+\left( s-k\right) _+^2}<1\) for \(k>1\), implies (2.4).
In order to prove (2.5), let us notice that, recalling (2.6), we have
So, since the function \(sg'\left( \left( s^2-k\right) _+\right) \) is continuous in the interval \( \left\{ {s\ge 0| s^2>k}\right\} =\left( \sqrt{k}, +\infty \right) \) and
then there exists a constant \(c_k>0\) such that
which is the conclusion. \(\square \)
For any \(\delta >0\), let us define
and observe that
Next lemma relates the function \(\mathcal {G}_\delta \left( \left| \xi \right| \right) \) with \(H_{\frac{p}{2}}(\xi )\).
Lemma 2.11
Let \(\mathcal {G}_\delta \) be the function defined by (2.8) and \(H_{\frac{p}{2}}\) be the one defined in (2.1) with \(\lambda =\displaystyle \frac{p}{2}\). Then we have
for any \(\xi , \eta \in \mathbb {R}^n\).
Proof
If \( |\xi |<1+\delta \) and \(|\eta |<1+\delta \) there is nothing to prove. So we will assume that \(|\xi |>1+\delta \), and without loss of generality we may suppose that \( |\eta |\le |\xi |\). Since \(\mathcal {G}_\delta (t)\) is increasing, we have
Now, it can be easily checked that
In the first case, we have
while, in the second,
Therefore,
for every \(\xi , \eta \in \mathbb {R}^n\), which is (2.10). \(\square \)
Arguing as in [21, Lemma 2.1], we prove the following.
Lemma 2.12
Let \(0<\delta \le 1\) and \(p\ge 2\). Then the following inequalities hold
with constants \({\tilde{c}}_{p,\delta }\) and \(c_{p,\delta }<\displaystyle \frac{2}{p}\) depending on p and \(\delta \).
Proof
If \(p=2\), one can explicitely calculate
and using the inequality \(\displaystyle \sqrt{a^2+b^2}\ge \frac{1}{\sqrt{2}}(a+b)\) with \(a=1+\delta \) and \(b=t\), we easily get
Let \(p>2\). The right inequality is a simple consequence of the trivial bound \(\displaystyle \frac{s}{\sqrt{1+\delta +s^2}}< 1\). For the left inequality we start observing that
Now, we calculate the integral in previous formula. By the change of variable \(r=\sqrt{1+\delta }+s\), we get
since \(0<\delta \le 1\), we have \(\delta \le \sqrt{1+\delta }\) and therefore \(r-\sqrt{1+\delta }+\delta \le r\). Calculating the last integral in previous formula, we get
Therefore the lemma will be proven if there exists a constant \(c_{p,\delta }<\displaystyle \frac{2}{p}\) such that
which, setting
is equivalent to prove that there exists \(c_{p,\delta }\) such that
It is easy to check that h(t) attains his maximum for \(t+\delta =\displaystyle \frac{2\sqrt{1+\delta }}{2-pc_{p,\delta }}\) and so
Therefore, to complete the proof it’s enough to solve the following equation
which is equivalent to
that, for \(0<\delta <1\), admits a unique solution \(c_{p,\delta }<\displaystyle \frac{2}{p}\). \(\square \)
3 The regularization
Let us fix \(z_0=(x_0,t_0)\in \Omega _T\) and \(R\in (0,1)\) such that \(Q_R(z_0)\Subset \Omega _T\). For \(\varepsilon >0\), we introduce the sequence of operators
and by
we denote the unique solution to the corresponding problems
where \(f^\varepsilon =f*\rho _\varepsilon \) with \(\rho _\varepsilon \) the usual sequence of mollifiers. One can easily check that the operator \(A_\varepsilon \) satisfies p-growth and p-ellipticity conditions with constants depending on \(\varepsilon \). Therefore, by the results in [19], we have
and, by the definition of \(V_{1, p}(\xi )\), this yields
By virtue of [3, Theorem 1.1], we also have \( H_{\frac{p}{2}}\left( Du^\varepsilon \right) \in L^2_{\textrm{loc}}\left( 0,T; W^{1,2}_\textrm{loc}\left( \Omega ,\mathbb {R}^n\right) \right) \) and, by the definition of \(H_{\frac{p}{2}}(\xi )\), it follows
3.1 Uniform a priori estimates
The first step in the proof of Theorem 1.1 is the following estimate for solutions to the regularized problem (3.2).
Lemma 3.1
Let \(u^\varepsilon \in C^0\left( t_0-{R^2},t_0;L^2\left( B_{{R}}\right) \right) \cap L^p\left( t_0-{R^2},t_0; u+W_0^{1,p}\left( B_{{R}}\right) \right) \) be the unique solution to (3.2). Then the following estimate
holds for any \(\varepsilon \in (0, 1]\) and for every \({Q_\rho \Subset Q_{2\rho }\Subset Q_R}\), with a constant \(c=c(n,p)\) independent of \(\varepsilon \).
Proof
The weak formulation of (3.2) reads as
for any test function \(\varphi \in C^{\infty }_0\left( Q_{{R}}\right) \). Recalling the notations introduced at (2.1) and (3.1), and replacing \(\varphi \) with \(\Delta _{-h}\varphi =\displaystyle \frac{\tau _{-h}\varphi }{h}\) for a sufficiently small \(h\in \mathbb {R}\setminus \left\{ {0}\right\} \), by virtue of the properties of difference quotients, we have
Arguing as in [19, Lemma 5.1], from (3.6) we get
For \(\Phi \in W^{1,\infty }_0\left( Q_{{R}}\right) \) non-negative and \(g\in W^{1,\infty }\left( \mathbb {R}\right) \) non-negative and non-decreasing, we choose \(\varphi =\Phi \cdot \Delta _h u^\varepsilon \cdot g\left( \left| \Delta _h u^\varepsilon \right| ^2\right) \) in the previous identity, thus getting
i.e.
that we rewrite as follows
Arguing as in [7, 8], the first integral in Eq. (3.7) can be expressed as follows
Using Lemma 2.2, since \(\Phi ,g\) are non-negative, we have
The right inequality in the assertion of Lemma 2.4 yields
Moreover, again by Lemmas 2.2 and 2.4 and the fact that \(g'(s)\ge 0\), we infer
Therefore (3.7) implies
Now let us consider a parabolic cylinder \(Q_{{\rho }}\left( z_0\right) \Subset Q_{{2\rho }}\left( z_0\right) \Subset Q_{{R}}\left( z_0\right) \) with \(\rho<{2\rho }<{R}\) and \(t_0>0\). For a fixed time \(\tau \in \left( t_0-{4\rho ^2}, t_0\right) \) and \(\theta \in \left( 0, t_0-\tau \right) \), we choose \(\Phi (x,t)=\eta ^2(x)\chi (t){\tilde{\chi }}(t)\) with \(\eta \in C_0^\infty \left( B_{{2\rho }}\left( x_0\right) \right) \), \(0\le \eta \le 1\), \(\chi \in W^{1,\infty }\left( \left[ 0,T\right] \right) \) with \(\partial _t\chi \ge 0\) and \({\tilde{\chi }}\) a Lipschitz continuous function defined, for \(0<\tau<\tau +\theta <T\), as follows
so that (3.8) yields
where we used the notation \(Q^\tau =B_{{2\rho }}\left( x_0\right) \times \left( t_0-{4\rho ^2},\tau \right) .\)
Since \(g\in W^{1,\infty }\left( \left[ 0, \infty \right) \right) \), by (3.3), by the last assertion of Lemma 2.9 and by Fatou’s Lemma, we have
and
Now let us observe that
and, using Hölder’s inequality with exponents \(\left( \frac{2\left( p-1\right) }{p-2}, \frac{2\left( p-1\right) }{p}\right) \), we have
and since, by (3.4), the right hand side of previous inequality is finite, again by Lemma 2.9 we have
which, since \(\Delta _{h}u^\varepsilon \rightarrow Du^\varepsilon \) strongly in \(L^2\left( 0, T; L^p\left( B_R\right) \right) \) as \(h\rightarrow 0, \) implies
Using similar arguments, we can check that
Now, by Proposition 2.7 (c), it holds
and choosing g such that
for a positive constant M and for every \(s>0\), we have
Since \(\Delta _{h}u^\varepsilon \rightarrow Du^\varepsilon \), \(\Delta _{-h}\left( \Delta _h u^\varepsilon \right) \rightarrow D^2u^\varepsilon \), \(\Delta _{-h}Du^\varepsilon \rightarrow D^2u^\varepsilon \) strongly in \(L^{2}\left( 0,T; L^2_\textrm{loc}\right. \left. \left( \Omega \right) \right) \) as \(h\rightarrow 0\), and \(f^\varepsilon \in C^\infty \left( \Omega _T\right) \), thanks to (3.16), we have
So, collecting (3.10), (3.11), (3.13), (3.14) and (3.17), we can pass to the limit as \(h\rightarrow 0\) in (3.9), thus getting
for every \(g\in W^{1,\infty }([0,+\infty ))\) such that (3.15) holds true. Now, by (3.12) and by Young’s inequality, we have
where we used (3.3), and where \(\sigma >0\) is a parameter that will be chosen later.
Now, using Young’s Inequality, we estimate the term \({\tilde{I}}_3\) as follows
Plugging (3.19) and (3.20) into (3.18), we get
which, for a sufficiently small \(\sigma \), gives
that, neglecting the third integral in the left hand side, implies
Now, for \(\delta \in (0,1)\), recalling the notation in (2.3), we choose
that is legitimate since, with this choice, \(g\in W^{1,\infty }([0,+\infty ))\) and \(g'(s)\ge 0.\)
Moreover, we have \(g(s)\in [0,1]\), for every \(s\ge 0\), and thanks to (2.5), there exists a constant \(c_\delta >0\) such that
so that (3.15) holds. Therefore, since g(s) vanishes on the set where \(s\le 1+\delta \) and \(g(s)\le 1\) for every s, (3.21) becomes
where we used that
since \(\delta <1\) and \(p\ge 2\). Using Young’s inequality in the first integral in the right hand, previous estimate yields
Choosing \(\beta \) sufficiently small, reabsorbing the second integral in the right hand side by the left hand side and using that \(g(s)\le 1\), we get
We now estimate the first integral in the right hand side of previous inequality with the use of (2.4) with \(s=\left| Du^\varepsilon \right| ^2\), \(A={\displaystyle \frac{\left( \left| Du^\varepsilon \right| -1\right) _+^{\frac{p}{2}}}{\left| Du^\varepsilon \right| }}\left| D^2 u^\varepsilon \right| \), \(B={\displaystyle \frac{c_p}{\delta ^{\frac{p}{2}}}}\left| f^\varepsilon \right| \) and \(k=1+\delta \), thus getting
where we also used the fact that \(\delta <1.\)
Inserting (3.23) in (3.22), we find
Choosing \(\alpha =\displaystyle \frac{1}{4}\), we can reabsorb the first integral in the right hand side by the left hand side, thus obtaining
By the definition of g, we have
and so it is easy to check that
that is
Therefore, by the previous equality and the properties of \(\chi \) and \(\eta \), (3.24) implies
which holds for almost every \(\tau \in \left( t_0-{4\rho ^2}, t_0\right) \).
We now choose a cut-off function \(\eta \in C^{\infty }\left( B_{{2\rho }}\left( x_0\right) \right) \) with \(\eta \equiv 1\) on \(B_\rho \left( x_0\right) \) such that \(0\le \eta \le 1\) and \(\left| D\eta \right| \le \displaystyle \frac{c}{{\rho }}.\) For the cut-off function in time, we choose \(\chi \in W^{1, \infty }\left( t_0-{R^2}, t_0, \left[ 0, 1\right] \right) \) such that \(\chi \equiv 0\) on \(\left( t_0-{R^2}, t_0-{4\rho ^2}\right] \), \(\chi \equiv 1\) on \(\left[ t_0-\rho ^2, t_0\right) \) and \(\partial _t\chi \le \displaystyle \frac{c}{{\rho ^2}}\) on \(\left( t_0-{4\rho ^2}, t_0-\rho ^2\right) \). With these choices, (3.25) gives
and, since \(\rho<2\rho<R<1\) and \(Q_{{2\rho }}=B_{\rho }\times \left( t_0-{4\rho ^2}, t_0\right) \), we have
Now, with \(\mathcal {G}_\delta (t)\) defined at (2.8), recalling (2.9), we have
One can easily check that \(g(s)\le g\left( s^2\right) \), and therefore
where we also used that \(g(s)=0\), for \(0<s\le 1+\delta .\) Using (3.27) in the left hand side of (3.26), we obtain
which is (3.5). \(\square \)
Combining Lemmas 3.1 and 2.8, we have the following.
Corollary 3.2
Let \(u^\varepsilon \in C^0\left( t_0-R^2,t_0;L^2\left( B_R\right) \right) \cap L^p\left( t_0-R^2,t_0; u+W_0^{1,p}\left( B_{R}\right) \right) \) be the unique solution to (3.2). Then the following estimate
holds for \(\left| h\right| <\displaystyle \frac{\rho }{4}\), for any parabolic cylinder \(Q_{2\rho }\Subset Q_R\left( z_0\right) \).
4 Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1, that will be divided in two steps.
In the first one we shall establish an estimate that will allow us to measure the \(L^2\)-distance between \(H_{\frac{p}{2}}\left( Du\right) \) and \(H_{\frac{p}{2}}\left( Du^\varepsilon \right) \) in terms of the \(L^2\)-distance between f and \(f^\varepsilon \).
In the second one, we conclude combining this comparison estimate with the one obtained for the difference quotient of the solution to the regularized problem at (3.28).
Proof of Theorem 1.1
Step 1: the comparison estimate.
We formally proceed by testing Eqs. (1.1) and (3.2) with the map \(\varphi =k(t)\left( u^\varepsilon -u\right) \), where \(k\in W^{1,\infty }\left( \mathbb {R}\right) \) is chosen such that
with \(t_{0}-{R^{2}}<t_{2}<t_{2}\,+\,\omega <t_{0}\), and then letting \(\omega \rightarrow 0\). We observe that, at this stage, it is important that \(u^\varepsilon \) and u agree on the parabolic boundary \(\partial _{\textrm{par}}Q_{{R}}\left( z_{0}\right) \).
Proceeding in a standard way (see for example [19]), for almost every \(t_{2}\in \left( t_{0}\,-\,{R^{2}},t_{0}\right) \), we find
where we used the abbreviation \(Q_{{R},t_{2}}=B_{{R}}\left( x_{0}\right) \times \left( t_{0}-{R^{2}},t_{2}\right) \). Using Lemma 2.1, the Cauchy-Schwarz inequality as well as Young’s inequality, from (4.1) we infer
where we set \(\lambda _p=\min \left\{ {\displaystyle \frac{1}{2}, \displaystyle \frac{4}{p^2}}\right\} .\) Reabsorbing the last integral in the right-hand side of (4.2) by the left-hand side, we arrive at
Using in turn Hölder’s inequality and Lemma 2.5, we get
Now, let us notice that
Inserting (4.5) in (4.4), we get
and, by Young’s inequality, we get
Inserting (4.6) in (4.3), we obtain
Choosing \(\displaystyle \beta =\frac{1}{2}\) and neglecting the third non-negative term in the left hand side of (4.7), we get
For further needs, we also record that, combining (4.5) and (4.8), we have
Step 2: The conclusion.
Let us fix \(\rho >0\) such that \(Q_{2\rho }\subset Q_R\). We start observing that
We estimate the right hand side of previous inequality using (3.28) and (2.10), as follows
that, thanks to (4.8), implies
Now, using (4.9), we get
which, combined with (4.10), implies
Taking the limit as \(\varepsilon \rightarrow 0\), and since \(f^\varepsilon \rightarrow f\) strongly in \(L^2\left( B_R\right) \), we obtain
and thanks to Lemma 2.9, we have \(\mathcal {G}_\delta \left( \left( \left| Du\right| -\delta -1\right) _+\right) \in L^2\left( t_0-{\rho ^2},t_0; W^{1,2}\left( B_{\rho }\right) \right) \) with the following estimate
Since previous estimate holds true for any \(\rho >0\) such that \(2\rho <R\), we may choose \(\rho =\displaystyle \frac{R}{4}\) thus getting (1.2). \(\square \)
5 Proof of Theorem 1.2
The higher differentiability result of Theorem 1.1 allows us to argue as in [19, Lemma 5.3] and [24, Lemma 3.2] to obtain the proof of Theorem 1.2.
Proof of Theorem 1.2
We start observing that
where \(c\equiv c(n,p)>0\) and \(\mathcal {G}_\delta (t)\) is the function defined at (2.8).
With the notation used in the previous sections, for \(B_{{2\rho }}\left( x_{0}\right) \Subset B_R\left( x_0\right) \), let \(\varphi \in C_{0}^{\infty }\left( B_{\rho }\left( x_{0}\right) \right) \) and \(\chi \in W^{1,\infty }\left( (0,T)\right) \) be two non-negative cut-off functions with \(\chi (0)=0\) and \(\partial _{t}\chi \ge 0\). Now, we fix a time \(t_{0}\in (0,T)\) and apply the Sobolev embedding theorem on the time slices \(\varSigma _{t}:=B_{\rho }(x_{0})\times \left\{ t\right\} \) for almost every \(t\in (0,t_{0})\), to infer that
where, in the second to the last line, we have applied Minkowski’s and Young’s inequalities one after the other. We estimate \(I_{1}(t)\) and \(I_{2}(t)\) separately. Let us first consider \(I_{1}(t)\). Using (5.1), Lemma 2.12 and Hölder’s inequality with exponents \(\displaystyle \left( \frac{n+2}{n}, \frac{n+2}{2}\right) \), we deduce
We now turn our attention to \(I_{2}(t)\). Lemma 2.12 and Hölder’s inequality yield
Putting together the last three estimates, using Lemma 2.12 in the left hand side, and integrating with respect to time, we obtain
where we have used the abbreviation \(Q^{t_{0}}:=B_{\rho }\left( x_{0}\right) \times \left( 0,t_{0}\right) \).
Now we choose \(\chi \in W^{1,\infty }\left( (0,T)\right) \) such that \(\chi \equiv 0\) on \(\left( 0,t_{0}-\rho ^{2}\right] \), \(\chi \equiv 1\) on \(\left[ t_{0}-\left( \displaystyle \frac{\rho }{2}\right) ^{2},T\right) \) and \(\partial _{t}\chi \ge 0\). For \(\varphi \in C_{0}^{\infty }\left( B_{\rho }\left( x_{0}\right) \right) \), we assume that \(\varphi \equiv 1\) on \(B_{\frac{\rho }{2}}\left( x_{0}\right) \), \(0\le \varphi \le 1\) and \(\left| D\varphi \right| \le \displaystyle \frac{C}{\rho }\).
With these choices (5.2) turns into
We now use (3.5), in order to estimate the integrals on the right-hand side of (5.3), thus getting
Estimate (4.9) implies
Let us observe that estimate (4.8) in particular yields
By the strong convergence of \(f^\varepsilon \rightarrow f\) in \(L^2\left( Q_R\right) \), passing to the limit as \(\varepsilon \rightarrow 0\), from previous estimate we deduce
that is \(H_{\frac{p}{2}}\left( Du^{\varepsilon }\right) \rightarrow H_{\frac{p}{2}}\left( Du\right) \), strongly in \(L^2\left( Q_{{R}}\right) .\) Therefore, up to a not relabelled subsequence, we also have \(H_{\frac{p}{2}}\left( Du^{\varepsilon }\right) \rightarrow H_{\frac{p}{2}}\left( Du\right) \), a.e. in \(Q_{{R}}\left( z_0\right) \) and so
By Fatou’s Lemma, taking the limit as \(\varepsilon \rightarrow 0\) in both sides of (5.4)
which holds for any \(\delta \in (0, 1)\), so fixing \(\delta =\displaystyle \frac{1}{2}\), we get the conclusion (1.3). \(\square \)
References
Akhmedov, Z.M., Barenblatt, G.I., Entov, V.M., Mirzadzhan-Zade, A.K.: Nonlinear effects in gas filtration. Fluid Dyn. 4(5), 68–72 (1969). https://doi.org/10.1007/BF01015960
Ambrosio, P.: Besov regularity for a class of singular or degenerate elliptic equations. J. Math. Anal. Appl. 505(2), 125636 (2022). https://doi.org/10.1016/j.jmaa.2021.125636
Ambrosio, P., Passarelli di Napoli, A.: Regularity results for a class of widely degenerate parabolic equations. Adv. Calc. Var. (2023). https://doi.org/10.1515/acv-2022-0062
Beck, L., Mingione, G.: Lipschitz bounds and nonuniform ellipticity. Commun. Pure Appl. Math. 73(5), 944–1034 (2020). https://doi.org/10.1002/cpa.21880
Bella, P., Schaffner, M.: Lipschitz bounds for integral functionals with \((p, \)g)-growth conditions. Adv. Calc. Var. (2022). https://doi.org/10.1515/acv-2022-
Bogelein, V., Duzaar, F., Giova, R., Passarelli di Napoli, A.: Higher regularity in congested traffic dynamics. Math. Ann. (2022). https://doi.org/10.1007/s00208-022-02375-y
Bogelein, V., Duzaar, F., Marcellini, P.: Parabolic equations with \(p, \)q-growth. J. Math. Pures Appl. 100(4), 535–563 (2013). https://doi.org/10.1016/j.matpur.2013.01.012
Bogelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with p, q-growth: a variational approach. Arch. Ration. Mech. Anal. 210(1), 219–267 (2013). https://doi.org/10.1007/s00205-013-0646-4
Brasco, L.: Global \(\text{ L}^{\infty }\) gradient estimates for solutions to a certain degenerate elliptic equation. Nonlinear Anal.: Theory Methods Appl. 74(2), 516531 (2011). https://doi.org/10.1016/j.na.2010.09.006
Brasco, L., Carlier, G., Santambrogio, F.: Congested traffic dynamics, weak flows and very degenerate elliptic equations. J. Math. Pures Appl. 93(6), 652–671 (2010). https://doi.org/10.1016/j.matpur.2010.03.010
Brasco, L., Santambrogio, F.: A sharp estimate a la Calderon–Zygmund for the p-Laplacian. Commun. Contemp. Math. 20(03), 1750030 (2018). https://doi.org/10.1142/S0219199717500304
Campanato, S.: Equazioni paraboliche del secondo ordine e spazi \(C^{2}{^{\prime }}^{e}\)(Q, 5). Ann. Mat. Pura Appl. 73(1), 55–102 (1966). https://doi.org/10.1007/BF02415082
Cianchi, A., Maz’ya, V.G.: Optimal second-order regularity for the p-Laplace system. J. Math. Pures Appl. 132, 41–78 (2019). https://doi.org/10.1016/J.MATPUR.2019.02.015
Cianchi, A., Maz’ya, V.G.: Second-Order two-sided estimates in nonlinear elliptic problems. Arch. Ration. Mech. Anal. 229(2018), 569–599 (2019). https://doi.org/10.1007/s00205-018-1223-7
Cupini, G., Giannetti, F., Giova, R., Passarelli di Napoli, A.: Regularity results for vectorial minimizers of a class of degenerate convex integrals. J. Differ. Equ. 265(9), 4375–4416 (2018). https://doi.org/10.1016/j.jde.2018
De Filippis, C., Mingione, G.: Lipschitz bounds and nonautonomous integrals. Arch. Ration. Mech. Anal. 242(2021), 973–1057 (2018). https://doi.org/10.1007/s00205-021-01698-5
DiBenedetto, E.: C\(^{\rm l+a}\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal.: Theory Methods Appl. 7(8), 827–850 (1983). https://doi.org/10.1016/0362-546X(83)90061-5
DiBenedetto, E.: Degenerate Parabolic Equations. Springer, Berlin (1993)
Duzaar, F., Mingione, G., Steffen, K.: Parabolic Systems with Polynomial Growth and Regularity. American Mathematical Society, Providence (2011)
Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz estimates for systems with ellipticity conditions at infinity. Ann. Mat. Pura Appl. (4) 195(5), 15751603 (2016). https://doi.org/10.1007/s10231-015-0529-4
Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. 13(3), 279–300 (2020). https://doi.org/10.1515/acv-2017-0037
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)
Lindqvist, P.: Notes on the Stationary p-Laplace Equation. Springer, Berlin (2019)
Scheven, C.: Regularity for subquadratic parabolic systems: higher integrability and dimension estimates. Proc. R. Soc. Edinb.: Sect. A Math. 140(6), 1269–1308 (2010). https://doi.org/10.1017/S030821050900167X
Acknowledgements
The work of the authors is supported by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). The authors have been also supported by the Universitá degli Studi di Napoli “Federico II” through the project FRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI.
Funding
Open access funding provided by Università Politecnica delle Marche within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose. The authors have no conflicts of interest to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.
Additional information
Communicated by Jan Kristensen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gentile, A., Passarelli di Napoli, A. Higher regularity for weak solutions to degenerate parabolic problems. Calc. Var. 62, 225 (2023). https://doi.org/10.1007/s00526-023-02564-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-023-02564-w