On the second-order regularity of solutions to the parabolic p-Laplace equation

In this paper, we study the second-order Sobolev regularity of solutions to the parabolic p-Laplace equation. For any p-parabolic function u, we show that D(Dup-2+s2Du)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(\left| Du\right| ^{\frac{p-2+s}{2}}Du)$$\end{document} exists as a function and belongs to Lloc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}_{\text {loc}}$$\end{document} with s>-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>-1$$\end{document} and 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document}. The range of s is sharp.


Introduction
The elliptic p-Laplace equation has an extensive literature on the second-order regularity. In contrast, the second-order regularity for the parabolic p-Laplace equation is much less studied. Throughout the paper, we have 1 < p < ∞. In the elliptic case, one of the known estimates shows W 1,2 loc regularity for the nonlinear expression of the gradient |Du| p−2+s 2 Du proven by Dong et al. [10] with s > 2 − min{ p + n n−1 , 3 + p−1 n−1 }, and then extended to s > −1 − p−1 n−1 by the third author [20]. The aim of this paper is to prove such a result to the parabolic p-Laplace equation. In other words, we prove in Theorem 2.2 that for any weak or viscosity solution u to (1.1), D(|Du| p−2+s 2 Du) exists, belongs to L 2 loc whenever s > −1, and this range is sharp.
In the parabolic case, Dong et al. [10] proved for p ∈ (1, 3) that the weak or viscosity solution u to (1.1) locally belongs to W 2,2 . This result is obtained as a special case from ours by selecting s = 2 − p. Our result also contains as a special case Lindqvist's [16] result for |Du| p−2 Du and |Du| The heuristic idea of the proof is to differentiate the equation (1.1), choose a test function ϕ = |Du| s u x k φ 2 and use a fundamental inequality (the name stems from [10] for a related inequality) from [20], which holds for any smooth function. Here u := n i=1 u x i x i denotes the Laplacian, ∞ u := n i, j=1 u x i x j u x i u x j the infinity Laplacian, and |D 2 u| := ( n i, j=1 u 2 x i x j ) 1/2 the Hilbert-Schmidt norm for matrices. Surprisingly, it is sufficient for the sharp result to use the previous inequality in a rather simple form which we obtain by an elementary fact that the square (|Du| 2 u − ∞ u) 2 is nonnegative. Naturally the fact that the form (1.2) is sufficient, simplifies the proof. At the same time, the form of (1.2) makes the coefficient C = C( p, s) of estimate (2.3) in Theorem 2.2 independent of n. Note that in the elliptic case, the fundamental inequalities in [10,20] include the parameter n so that both the range of s and C depend on n.
Unlike the second-order regularity, the lower-order regularity of the parabolic p-Laplace equation has been extensively studied since the 1980s, see DiBenedetto's monograph [6] as well as, for example [2,8,9,14,15,22,23]. In the elliptic case, the second-order Sobolev regularity has been studied in addition to above mentioned [10], for example, in [1,3,18], and for a different parabolic equation in [11].

Preliminaries and main results
Let x 0 ∈ R n , n ≥ 1 and r > 0. We denote by the usual Euclidean ball in R n . For a space-time point (x 0 , t 0 ) ∈ R n+1 and a radius r > 0, we define the parabolic cylinder as To ease the notation, we may write Q r := Q r (x 0 , t 0 ). Let ⊂ R n denote an open domain. For T > 0, we set If U is compactly contained in , i.e., U ⊂ and the closure of U is a compact subset of , we write U . For 0 < t 1 < t 2 < ∞, we set We denote the first partial derivatives of a function u : T → R by u x k and u t . The spatial gradient is denoted by Du, and the second derivatives by u x i x j . Further, D 2 u stands for matrix of second derivatives with respect to the space variables. As usual, the Sobolev space W 1, p (U ) denotes the space of measurable functions u such that u ∈ L p (U ) and the distributional first partial derivatives u x i exist in U and belong to L p (U ). We use the norm By the parabolic Sobolev space L p (t 1 , t 2 ; W 1, p (U )), with 0 < t 1 < t 2 < ∞, we mean the space of measurable functions u(x, t) such that the mapping x → u(x, t) belongs to W 1, p (U ) for almost every t 1 < t < t 2 and the norm where p u := div |Du| p−2 Du is the p-Laplace operator with 1 < p < ∞.
T is an open cylinder, we have u ∈ C(U t 1 ,t 2 ) ∩ L p (t 1 , t 2 ; W 1, p (U )), and u satisfies the integral equality Such solutions are called p-parabolic functions.
Under the above definition, weak solutions are equivalent to viscosity solutions to (2.1) for 1 < p < ∞, see [13,19,21]. In this setting, gradients are bounded and Hölder continuous by a recent work of Imbert et al. [12]. Their result covers a more general class of equations containing the parabolic p-Laplace equation and the normalized p-parabolic equation arising from the game theory [17]. For earlier C 1,α -regularity results based on the variational approach with various assumptions, see DiBenedetto and Friedman [7], Wiegner [24], Chen [4], and Chapter IX in [6].

Main results
For s ∈ R, we define the vector field V s : R n → R n as where C = C( p, s) > 0 and Q r ⊂ Q 2r T are concentric parabolic cylinders.
Note that here the range of s and the coefficient C do not depend on n. For the parabolic case, the range of s must satisfy the constraints of both the elliptic and parabolic terms. We get the elliptic restriction s > −1 − p−1 n−1 by [20] and s > −1 rising from the parabolic terms (see Remark 2.4). By combining them, we get the Remark 2.3. In particular, we may set s = 0, and s = p − 2 for any 1 < p < ∞ reproving Lindqvist's result in [16] for |Du| p−2 2 Du and |Du| p−2 Du. If 1 < p < 3, we may set s = 2 − p to reprove the second-order Sobolev regularity obtained in [10].
Remark 2.4 (Counter example). The counterexample from [10] turns out to work also in our case, and shows that the range s > −1 in Theorem 2.2 is sharp. By a direct calculation, the function if and only if s > −1. Indeed, we have and Then, Once we have proven the main result, Theorem 2.2, the existence and integrability of the time derivative easily follows as pointed out by Lindqvist [16] and Dong et al. [10]. We give the short proof for the convenience of the reader.
By the weak formulation we get that u t exists, and u t ∈ L 2 loc ( T ).

Idea of the proof
In this section, for the convenience of the reader, we present the formal idea of the proof without excess details. In this setting, we assume that u ∈ C ∞ ( T ) and Du = 0. The detailed proof is presented in Sect. 4.
Differentiating with respect to x k in (2.1), we get Here, I denotes the n × n identity matrix and Du ⊗ Du stands for the tensor product of two vectors in R n , resulting in a matrix in R n×n with the entries u x i u x j . We first study the term on the left hand side of (3.1) and choose a test function Now the first integral on the right hand side is of the same form as in the elliptic case, and thus, the proof of [20,Lemma3.3] gives that for any η > 0, where N ∞ u := |Du| −2 n i, j=1 u x i x j u x i u x j stands for the normalized or game theoretic infinity Laplacian. Observe that on the right hand side, we have bounded terms only. As a corollary, similarly as in [20,Corollary 3.4], we get Next we estimate the first term on the right hand side in (3.3). Using the inequality of [20, Corollary 2.2]: dividing both sides by |Du| 4 , we have On the last line, we used ( u − N ∞ u) 2 ≥ 0. Now we use the previous inequality in (3.2) for the term containing D 2 u 2 , set η = min{ 1 4 ( p + s), 1 6 ( p − 1)(s + 1)}, and obtain whenever s > −1. We also used and combining this with the previous estimate, we finally get The estimate in Theorem 2.2 is obtained by choosing φ as a standard cutoff function.

Detailed proof
In this section, we present a detailed proof of Theorem 2.2 by regularizing Eq. (2.1). Solutions to the regularized equation will be smooth, and thus, the differentiation of this equation is justified. Since the obtained estimates will be uniform with respect to the regularization, we will be able to pass to the original equation at the end.
To start with the above plan, let u : T → R be a p-parabolic function. Fix a smooth subdomain U and 0 < t 1 < t 2 < ∞ such that U t 1 ,t 2 T . Let > 0 be small and u : U t 1 ,t 2 → R be a weak solution to where μ := |Du | 2 + and the parabolic boundary is defined as According to standard parabolic theory, we get u ∈ C ∞ (U t 1 ,t 2 ) ∩ C(U t 1 ,t 2 ), see [7,24].

Lemma 4.1. Let u : U t 1 ,t 2 → R be a weak solution to (4.1). If s > −1, then for any
To prove Lemma 4.1, we use the inequality (1.2). In Sect. 3, under the assumption Du = 0, we can directly divide both sides of (1.2) by |Du| 4 to get the inequality (3.4), which gives the lower bound of D 2 u 2 . In order to get an inequality similar to (3.4), we also need to consider the case when Du = 0. Thus, we reformulate (1.2) here in a way that allows us to apply it in this context. For the reformulation, we introduce some notations. Let v : U t 1 ,t 2 → R be a smooth function. In particular, |Dv| is locally Lipschitz continuous (by triangle inequality) and thus, by Rademacher's theorem, differentiable almost everywhere on each time slice, hence also in U t 1 ,t 2 .
Note (x 0 , t 0 )) such that |Dv|(ξ ) < 0, which is obviously impossible. On the other hand, if Dv(x 0 , t 0 ) = 0 for some (x 0 , t 0 ) ∈ U t 1 ,t 2 , then |Dv| is differentiable at (x 0 , t 0 ) and For each point in U t 1 ,t 2 where Dv = 0, we fix an orthonormal basis of R n , {e 1 , . . . , e n }, such that e n = Dv |Dv| . Hence, we have, for those points where Dv = 0, For those points where |Dv| is differentiable, let us define the part of D|Dv| which is tangential to the spatial level sets of v as We employ these notations to write In Sect. 3, without dividing |D|Dv|| 2 into two parts, we use the inequality (3.4) and (3.6) to get the estimate (3.5). When using (3.6), we need to be careful and check that if the coefficient of |D|Dv|| 2 is nonnegative. For the regularization, the coefficients of each terms become more complicated, and thus, by using the equality (4.2), we can consider the coefficients together in the last step of the estimate. Now we can restate (1.2).

Lemma 4.2.
Let v : U t 1 ,t 2 → R be a smooth function. Then, Proof. Recall that |Dv| is differentiable a.e. in U t 1 ,t 2 . From now on, consider such points of U t 1 ,t 2 where |Dv| is differentiable.
Dividing both sides by |Dv| 4 , using the definitions of D T |Dv| and N ∞ v, we get the desired inequality by following: Proof of Lemma 4.1. The spatial partial derivatives u x k , k = 1, . . . , n, solve Note that min{1, p − 1}I ≤ A ≤ max{1, p − 1}I (4.5) uniformly in U t 1 ,t 2 and for .
We multiply Eq. (4.4) with μ s u x k , where s > −1, and obtain For the first item in the above display, we note that Summing (4.6) over k = 1, . . . , n and taking (4.7) into account gives that (4.9) Above we used and a straightforward computation. In other words, By (4.8) and (4.10), we have We claim that for s > −1, we can find a small number λ = λ( p, s) > 0 such that Observe that this is not a trivial inequality since not all the coefficients are positive in σ . If (4.12) holds, then the desired estimate follows easily. Indeed, we plug the estimate (4.12) into Eq. (4.11) to obtain . Multiplying (4.13) by φ 2 and then integrating over U t 1 ,t 2 yields We employ integration by parts, (4.5) and Young's inequality to obtain the upper bound of the right hand side term in above inequality, for any η > 0 and some constant C = C( p) > 0. The desired estimate follows by choosing η = λ 2 . It remains to prove (4.12). As explained above in this section, we can write almost everywhere in U t 1 ,t 2 . For λ ∈ (0, 1), we write (4.14) For the latter part of σ on the right hand side of (4.14), we utilize the nonnegativity of the |D 2 u | 2 -term via the inequality of Lemma 4.2: We obtain a lower bound allows us to divide the terms in σ and τ according to the degree of . This kind of regrouping is useful, because it separates the main terms that appear also in the formal calculation of Sect. 3 from those terms that appear as a result of the regularization. Indeed, we write (4.17) As we plug (4.16) and (4.17) into (4.15), we can easily choose λ = λ( p, s) > 0 so small that This is indeed possible because s > −1. Now that we have shown (4.12), the proof is finished.
Proof of Theorem 2.2. To prove Theorem 2.2, we need to justify letting → 0 in  for any φ ∈ C ∞ 0 (U t 1 ,t 2 ). The estimate (2.3) can be derived from (4.18) as follows. Let us fix a space-time point (x 0 , t 0 ) ∈ U t 1 ,t 2 . Let r > 0 be small enough such that the parabolic cylinder with center (x 0 , t 0 ) and radius 2r fits inside U t 1 ,t 2 , that is Q 2r U t 1 ,t 2 . Let φ ∈ C ∞ 0 (U t 1 ,t 2 ) be a cutoff function such that φ ≡ 1 in Q r , |φ| ≤ 1, spt φ ⊂ Q 2r , |Dφ| ≤ 10 r and |φ t | ≤ The estimate (4.18) implies that Q r |D(V s (Du ))| 2 dxdt 2 dxdt (4.20) for C = C( p, s) > 0. Since s > −1, we can apply, for example, [12] to conclude for the gradient Thus Du converge uniformly (and strongly in L p ) by Arzelà-Ascoli theorem. It follows that the limit u is a solution to (2.1). Moreover, the right hand side of (4.20) is thus bounded from above by a constant independent of . Thus {D(V s (Du ))} is bounded in L 2 (Q r ), and consequently we may extract a subsequence that converges weakly in L 2 (Q r ). Further, using integration by parts, we see that the limit is D(V s (Du)), and thus which is the desired estimate.