Regularity of Solutions of the Parabolic Normalized p-Laplace Equation

The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.

In the linear case p = 2, we have the heat equation u t = ∆u, and also for n = 1, the equation reduces to the heat equation u t = (p − 1)u xx . At the limit p = 1, we obtain the equation for motion by mean curvature. We aim at showing that the time derivative ∂u ∂t exists in the Sobolev sense and belongs to L 2 loc (Ω T ). We also study the second derivatives ∂ 2 u ∂x i ∂x j . There has been some recent interest in connection with stochastic game theory, where the equation appears, cf. [7]. From our point of view, the work [3] is of actual interest, because there it is shown that the time derivative u t of the viscosity solutions exists and is locally bounded, provided that the lateral boundary values are smooth. Thus, the boundary values control the time regularity. If no such assumptions about the behaviour at the lateral boundary ∂Ω × (0, T) are made, a conclusion like u t ∈ L ∞ loc (Ω T ) is in doubt. Our main result is the following, where we unfortunately have to restrict p. Theorem 1.1. Suppose that u = u(x, t) is a viscosity solution of the normalized p-Laplace equation in Ω T . If 6 5 < p < 14 5 , then the Sobolev derivatives ∂u ∂t and ∂ 2 u ∂x i ∂x j exist and belong to L 2 loc (Ω T ).
We emphasize that no assumptions on the boundary values are made for this interior estimate. Our method of proof is based on a verification of the identity where we have to prove that the function U, which is the right-hand side of equation (1.1), belongs to L 2 loc (Ω T ). Thus, the second spatial derivatives D 2 u are crucial (local boundedness of ∇u was proven in [2,3] and interior Hölder estimates for the gradient in [6]). The elliptic case has been studied in [1].
In the range 1 < p < 2, one can bypass the question of second derivatives. Theorem 1.2. Suppose that u = u(x, t) is a viscosity solution of the normalized p-Laplace equation in Ω T . If 1 < p < 2, then the Sobolev derivative ∂u ∂t exists and belongs to L 2 loc (Ω T ).
To avoid the problem of vanishing gradient, we first study the regularized equation Here the classical parabolic regularity theory is applicable. The equation was studied by Does in [3], where an estimate of the gradient ∇u ϵ was found with Bernstein's method. We shall prove a maximum principle for the gradient. Further, we differentiate equation (1.2) with respect to the space variables and derive estimates for u ϵ , which are passed over to the solution u of (1.1). Analogous results seem to be possible to reach through the Cordes condition. This also restricts the range of valid exponents p. We have refrained from this approach, mainly since the absence of zero (lateral) boundary values produces many undesired terms to estimate. Finally, we mention that the limits 6 5 and 14 5 in Theorem 1.1 are evidently an artifact of the method. It would be interesting to know whether the theorem is valid in the whole range 1 < p < ∞. In any case, our method is not capable to reach all exponents.

Preliminaries
Notation. The gradient of a function f : and its Hessian matrix is We shall, occasionally, use the abbreviation is often referred to. Finally, the summation convention is used when convenient. Viscosity solutions. The normalized p-Laplace equation is not in divergence form. Thus, the concept of weak solutions with test functions under the integral sign is problematic. Fortunately, the modern concept of viscosity solutions works well. The existence and uniqueness of viscosity solutions of the normalized p-Laplace equation was established in [2]. We recall the definition. Definition 2.1. We say that an upper semi-continuous function u is a viscosity subsolution of equation (1.1) if for all ϕ ∈ C 2 (Ω T ), we have at any interior point (x, t) where u − ϕ attains a local maximum, provided ∇ϕ(x, t) ̸ = 0. Further, at any interior point (x, t) where u − ϕ attains a local maximum and ∇ϕ(x, t) = 0, we require for some η ∈ ℝ n , with |η| ≤ 1. Definition 2.2. We say that a lower semi-continuous function u is a viscosity supersolution of equation (1.1) if for all ϕ ∈ C 2 (Ω T ), we have For a detailed discussion on the definition at critical points, we refer to [5]. The reason behind the choice of η ∈ ℝ n is given in [5,Section 2]. The viscosity solutions of equation (1.2) are defined in a similar manner, except that now ∇ϕ(x, t) = 0 is not a problem. Maximum principle for the gradient. In order to estimate the time derivative, we need bounds on the second derivatives of u ϵ (and also on its gradient). If we first assume that u ϵ is C 1 on the parabolic boundary ∂ par Ω T , we get bounds on the gradient in all of Ω T . This follows from the following maximum principle.

Proposition 2.4 (Maximum principle). Let u ϵ be a solution of equation
Proof. With some modifications, a proof can be extracted from [3]. We give a direct proof. To this end, consider To find the partial differential equation satisfied by V ϵ , we calculate¹ Writing equation (1.1) in the form Rearranging and using ∆V ϵ = 2|D 2 u ϵ | 2 + 2⟨∇u ϵ , ∇∆u ϵ ⟩, we arrive at the following differential equation for V ϵ : Suppose that w ϵ has an interior maximum point at (x 0 , t 0 ). At this point, V ϵ (x 0 , t 0 ) > 0, otherwise we would have V ϵ (x, t) ≡ 0 in Ω T , in which case there is nothing to prove. By the infinitesimal calculus, where we have included the case t 0 = T. Further, the matrix D 2 w(x 0 , t 0 ) is negative semidefinite. Using equation (2.1) and noting that ∇w = ∇V ϵ and D 2 w = D 2 V ϵ , we get, at (x 0 , t 0 ), To avoid the contradiction α ≤ 0, w must attain its maximum on the parabolic boundary.
Hence, for any (x, t) ∈ Ω T , we have We finish the proof by sending α → 0 + .
With no assumptions for u ϵ on the parabolic boundary, we need a stronger result, taken from [3, p. 381].

Theorem 2.5. Let u ϵ be a solution of equation
Note that no condition on the lateral boundary ∂Ω × [0, T] was used. By continuity, follows. (Here one can pass to the limit as ϵ → 0.) The proof of the lemma below, a simple special case of the Miranda-Talenti lemma, can be found for smooth functions in [4, p. 308]. If f is not smooth, we perform a strictly interior approximation, so that no boundary integrals appear (which is possible since ξ ∈ C ∞ 0 ). Lemma 2.6 (Miranda-Talenti). Let ξ ∈ C ∞ 0 (Ω T ) and f ∈ L 2 (0, T, W 2,2 (Ω)). Then

Regularization
The next lemma tells us that the solutions of (1.2) converge locally uniformly to the viscosity solution of (1.1). Proof. By Theorem 2.5, we can use Ascoli's theorem to extract a convergent subsequence u ϵ j converging locally uniformly to some continuous function, namely, u ϵ j → v. We claim that v is a viscosity solution of equation (1.1). The lemma then follows by uniqueness.
Our proof of Theorem 1.1 consists in showing that the second derivatives D 2 u ϵ belong locally to L 2 with a bound independent of ϵ. Once this is established, we see that Hence, for any bounded subdomain D ⊂⊂ Ω T , with C independent of ϵ. By this uniform bound, there exists a subsequence such that, as j → ∞, In particular, this means that U ∈ L 2 (D) and for any ϕ ∈ C ∞ 0 (D), we have If u is the unique viscosity solution of (1.1), we invoke Lemma 3.1 and the calculations above to find, for any test function ϕ ∈ C ∞ 0 (D), This shows that the Sobolev derivative u t exists and, since the previous equation holds for any subdomain D ⊂⊂ Ω T , we conclude that ∂u ∂t = U ∈ L 2 loc (Ω T ). To complete the proof of Theorem 1.1, it remains to establish the missing local bound of ‖D 2 u ϵ ‖ L 2 uniformly in ϵ.
Differentiating equation (1.2) with respect to the variable x j , we obtain ∂ ∂t Take ξ ∈ C ∞ 0 (Ω T ), with ξ ≥ 0. Multiply both sides of the equation by ξ 2 V ϵ u ϵ j and sum j from 1 to n. Integrate over Ω T , using integration by parts and keeping in mind that ξ is compactly supported in Ω T , to obtain Writing out the derivatives gives the fundamental formula In the next section we shall bound the main term I uniformly with respect to ϵ.