Higher regularity for weak solutions to degenerate parabolic problems

In this paper, we study two related features of the regularity of the weak solutions to the following strongly degenerate parabolic equation ut-divDu-1+p-1DuDu=finΩT=Ω×(0,T),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t-\textrm{div}\left( \left( \left| Du\right| -1\right) _+^{p-1}\frac{Du}{\left| Du\right| }\right) =f\qquad \text{ in } \Omega _T =\Omega \times (0,T), \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a bounded domain in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{n}$$\end{document} for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, p≥andT>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge \text {and}\, T>0$$\end{document}. We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that f∈Lloc2ΩT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L^{2}_{\textrm{loc}}\left( \Omega _T\right) $$\end{document}. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f.


Introduction
In this paper, we study the regularity properties of weak solutions u : Ω T → 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,Section 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 |Du| ≤ 1 and hence its principal part of behaves like a p-Laplacian operator only at infinity.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 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 [9], by Duzaar et al. in [13] in case of superquadratic growth, while Scheven in [17] faced the subquadratic growth case.In the above mentioned papers, the problem have 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 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 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 Moreover, it is well known that in case of degenerate problems (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 (see, for example [8] for elliptic and [3] for parabolic equations).Actually, the sharp assumption for the datum in the elliptic setting has been determined in [8] as a fractional Sobolev regularity suitably related to the growth exponent p and the dimension n.
The main aim of this paper is to show that without assuming any kind of Sobolev regularity for the datum, but assuming only f ∈ L 2 , we are still able to obtain higher differentiability for the weak solutions but outside a set larger than the degeneracy set of the problem.It is worth mentioning 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, we shall prove the following Theorem 1.1.Let n ≥ 2, p ≥ 2 and f ∈ L 2 loc (Ω T ).Moreover, let us assume that u ∈ C 0 0, T ; L 2 (Ω) ∩ L p loc 0, T ; W 1,p loc (Ω) is a weak solution to (1.1).Then, for any δ ∈ (0, 1), we have where Moreover the following estimate 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 ( [11]).The major effort in the proof of previous Theorem 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 a suitable nonlinear function of the gradient of the solution that vanishes in the set { |Du| ≤ 1 + δ }, with δ > 0.
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 [10]).Anyway, as a consequence of the higher differentiability result in Theorem 1.1, we establish a higher integrability result for the spatial gradient of the solution to equation (1.1), which is the following Theorem 1.2.Under the assumptions of Theorem 1.1, we have with the following estimate The proof of previous Theorem consists in using an interpolation argument with the aim of establishing an estimate for the L p+ 4 n norm of the gradient of the solutions to the approximating problems that is preserved in the passage to the limit.We conclude mentioning 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 (see e.g.[2,4,6,8]).Moreover, we'd like to stress that, for sake of clarity, we confine ourselves to equation (1.1), but we believe that our techniques apply as well to a general class of equations with a widely degenerate structure.

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 R n will be the standard Euclidean one and it will be denoted by | • |.In particular, for the vectors ξ, η ∈ R n , we write ξ, η for the usual inner product and |ξ| := ξ, ξ 1 2 for the corresponding Euclidean norm.
For points in space-time, we will use abbreviations like z = (x, t) or z 0 = (x 0 , t 0 ), for spatial variables x, x 0 ∈ R n and times t, t 0 ∈ R. We also denote by the open ball with radius ρ > 0 and center x 0 ∈ R n ; when not important, or clear from the context, we shall omit to indicate the center, denoting: B ρ ≡ B (x 0 , ρ).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 (x 0 , t 0 ) and width ρ.We shall sometimes omit the dependence on the vertex when the cylinders occurring share the same vertex.Finally, for a cylinder , where A ⊂ R n and t 1 < t 2 , we denote by the usual parabolic boundary of Q, which is nothing but the standard topological boundary without the upper cap A × { t 2 }.
We now recall some tools that will be useful to prove our results.
For the auxiliary function H λ : R n → R n defined as where λ > 0 is a parameter, we record the following estimates (see [7,Lemma 4.1]): for every ϕ ∈ C ∞ 0 (Ω T ).In the following, we shall also use the well known auxiliary function V p : R n → R n defined as where p ≥ 2. We have the following result.
Lemma 2.4.For every ξ, η ∈ R n there hold We refer to [16,Chapter 12] or to [15,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 [12, Proposition 3.1] Lemma 2.5.Assume that the function v for some exponents 1 ≤ p, q < ∞ .Then the following estimate ˆQr(z0) |v| holds true for a positive constant c depending at most on n, p and q.

Difference quotients
We recall here the definition and some elementary properties of the difference quotients (see, for example, [15,Chapter 8]).
Definition 2.6.For every function F : R n → R N the finite difference operator in the direction x s is defined by where h ∈ R, e s is the unit vector in the direction x s and s ∈ {1, . . ., n}.
The difference quotient of F with respect to x s is defined for h ∈ R \ {0} as We shall omit the index s when it is not necessary, and simply write τ h F (x) = F (x + h) − F (x) and Proposition 2.7.Let F ∈ W 1,p (Ω), with p ≥ 1, and let us set Then: The next result about the finite difference operator is a kind of integral version of Lagrange Theorem (see [15,Lemma 8.1]).

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 ∈ [0, +∞), 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 ≥ 0 the following Young's type inequality holds for every parameters α, σ > 0 with a constant c α independent of σ.Moreover, there exists a constant c k > 0, depending on k, such that both the conclusions trivially hold for s ≤ √ k.Now assume that s > √ k and note that Young's inequality implies where we used the explicit expression of g ′ k (s) at (2.6).Recalling (2.3) and since (2.8) Inserting this in (2.8), we get (2.4).
In order to prove (2.5), let us notice that, recalling (2.6), we have So, since the function sg which is the conclusion.
For any δ > 0, let us define and observe that for any ξ, η ∈ R n .
Proof.If |ξ| < 1 + δ and |η| < 1 + δ there is nothing to prove.So will assume that |ξ| > 1 + δ, and without loss of generality we may suppose that |η| ≤ |ξ|.Since G δ (t) is increasing, we have . Now, it can be easily checked that In the first case, we have while, in the second, Therefore, Arguing as in [14, Lemma 2.1], we prove the following.
Proof.If p = 2, one can easily calculate Let p > 2. The right inequality is a simple consequence of the trivial bound s √ 1+δ+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 = √ 1 + δ + s, we get Calculating the last integral in previous formula, we get Therefore the lemma will be proven if there exists a constant c p,δ < 2 p such that which, setting is equivalent to prove that there exists c p,δ such that It is easy to check that h(t) attains his maximum for t and so Therefore, to complete the proof it's enough to solve the following equation which is equivalent to that, for 0 < δ < 1, admits a unique solution c p,δ < 2 p .

The regularization
For ε > 0, we introduce the sequence of operators and by we denote the unique solution to the corresponding problems where Q R (z 0 ) ⋐ Ω T with R < 1, f ε = f * ρ ε with ρ ε the usual sequence of mollifiers.One can easily check that the operator A ε satisfies p-growth and p-ellipticity assumptions with constants depending on ε.Therefore, by the results in [13], we have and, by the definition of V p (ξ), this yields By virtue of [3, Theorem 1.1], we also have

Uniform a priori estimates
The first step in the proof of Theorem 1.1 is the following estimate for solutions to the regularized problem (3.1).
) be the unique solution to (3.1).Then the following estimate holds for any ε ∈ (0, 1] and for every Proof.The weak formulation of (3.1) reads as Recalling the notation used in (2.2), and replacing ϕ with ∆ −h ϕ = τ −h ϕ h for a sufficiently small h ∈ R \ { 0 }, by virtue of the properties of difference quotients, we have Arguing as in [13, Lemma 5.1], from (3.5) we get For Φ ∈ W 1,∞ 0 (Q R ) non negative and g ∈ W 1,∞ (R) non negative and non decreasing, we choose that we rewrite as follows Arguing as in [5],the first integral in equation (3.6) can be expressed as follows Using Lemma 2.2, since Φ, 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) ≥ 0, we infer For a fixed time τ ∈ t 0 − 4ρ 2 , t 0 and θ ∈ (0, ) with ∂ t χ ≥ 0 and χ a Lipschitz continuous function defined, for 0 < τ < τ + θ < T , as follows where we used the notation 2), by the last assertion of Lemma 2.9 and by Fatou's Lemma, we have lim inf and, using Hölder's inequality with exponents 2(p−1) p−2 , 2(p−1) p , we have , and since, by (3.3), the right hand side of previous inequality is finite again by Lemma 2.9, we have Using similar arguments, we can check that lim h→0 Now, by Proposition 2.7(c), it holds and choosing g such that sg ′ s 2 ≤ M, (3.14) for a positive constant M , we have So, collecting (3.9), (3.10), (3.12), (3.13) and (3.16), we can pass to the limit as h → 0 in (3.8), thus getting for every g ∈ W 1,∞ (0, +∞) such that (3.14) holds true.Now, by (3.11) and by Young's inequality, we have where we used (3.2), and where σ > 0 is a parameter that will be chosen later.Now, using Young's Inequality, we estimate the term Ĩ3 , as follows which, for a sufficiently small σ, gives that, neglecting the third integral in the left hand side, implies Now, for δ ∈ (0, 1), recalling the notation in (2.3), we choose Moreover, with this choice, we have g(s) ∈ [0, 1], for every s ≥ 0, and thanks to (2.5), there exists a constant c δ > 0 such that sg ′ s 2 ≤ c δ for every s ≥ 0, so that (3.14) holds.Therefore, since g(s) vanishes on the set where s ≤ 1 + δ and g(s) ≤ 1 for every s, (3.20) becomes where we used that sup , since δ < 1.Using Young's inequality in the first integral in the right hand, previous estimate yields Choosing β sufficiently small, reabsorbing the second integral in the right hand side by the left hand side and using that g(s) ≤ 1, we get We now estimate the first integral in the right side of previous inequality with the use of (2.4) with with constants c, c α both independent of σ and where we used that δ < 1.By virtue of (3.3), taking the limit as σ → 0 in previous inequality, we have Inserting (3.22) in (3.21), we find , 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 Therefore, by previous equality and the properties of χ and η, (3.23) implies which holds for almost every τ ∈ t 0 − 4ρ 2 , t 0 .We now choose a cut-off function and since ρ < 2ρ < R < 1, and Now, with G δ (t) defined at (2.9), recalling (2.10), we have Since g(s) is nondecreasing, we have g(s) ≤ g s 2 , and therefore where we also used that g(s) = 0, for 0 < s ≤ 1 + δ.Using (3.26) in the left hand side of (3.25), we obtain which is (3.4).
Combining Lemma 3.1 and Lemma 2.8, we have the following.
) be the unique solution to (3.1).Then the following estimate

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 p 2 (Du) and H p 2 (Du ε ) in terms of the L 2 -distance between f and f ε .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 (3.27).
Proof of Theorem 1.1.Step 1: the comparison estimate.We formally proceed by testing equations (1.1) and (3.1) with the map with t 0 − R 2 < t 2 < t 2 + ω < t 0 , and then letting ω → 0. We observe that, at this stage, it is important that u ε and u agree on the parabolic boundary Proceeding in a standard way (see for example [13]), for almost every t 2 ∈ t 0 − R 2 , t 0 , we find where we used the abbreviation Q R,t2 = B R (x 0 ) × t 0 − R 2 , t 2 .Using Lemma 2.1, the Cauchy-Schwarz inequality as well as Young's inequality, from (4.1) we infer where we set λ p = min 1 2 , 4 p 2 .Reabsorbing the last integral in the right-hand side of (4.2) by the left-hand side, we arrive at sup t∈(t0−R 2 ,t0) Using in turn Hölder's inequality and Lemma 2.5, we get and, by Young's inequality, we get Choosing β = 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 ρ > 0 such that Q 2ρ ⊂ Q R .We start observing that We estimate the right hand side of previous inequality using (3.27) and (2.11), as follows that, thanks to (4.8), implies Now, using (4.9), we get Taking the limit as ε → 0, and since f ε → f strongly in L 2 (B R ), we obtain and thanks to Lemma 2.9, we have Since previous estimate holds true for any ρ > 0 such that 4ρ < R, we may choose ρ = R 8 thus getting (1.2).

Proof of Theorem 1.2
The higher differentiability result of Theorem 1.1 allows us to argue as in [13,Lemma 5.3] and [17,Lemma 3.2] to obtain the proof of Theorem 1.2.
Proof of Theorem 1.2.We start observing that where c ≡ c(n, p) > 0 and G δ (t) is the function defined at (2.9).

p 2 ( 2 +
Du) = (|Du| − 1) p Du |Du| .It is worth noticing that, as it can be expected, this function of the gradient doesn't give information on the second 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 div (H p−1 (Du)) = 0, where H p−1 (Du) = (|Du| − 1) p−1 + Du |Du| , no more than Lipschitz regularity can be expected.