Gradient regularity for a singular parabolic equation in non-divergence form

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $$\partial_t u-|Du|^\gamma\Delta_p^N u=f,$$ where $-1<\gamma<0$, $1<p<\infty$, and $f$ is a given bounded function. We establish interior H\"older regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.


Introduction
We study gradient regularity of the following singular parabolic equation in non-divergence form, Here ∆ N p u := ∆u + (p − 2) D 2 u Du |Du| , Du |Du| is the normalized p-Laplacian. We assume that −1 < γ < 0, 1 < p < ∞, and f is a given continuous and bounded function.
When γ = p − 2, equation (1.1) is the standard parabolic p-Laplace equation and in that case it is possible to consider both distributional weak solutions and viscosity solutions. In the case of bounded weak solutions, equivalence with viscosity solutions was shown by Juutinen, Lindqvist, and Manfredi [25]. For that equation, Hölder regularity of the gradient was shown by DiBenedetto and Friedman [17] and Wiegner [36], see also Kuusi and Mingione [26] and references therein. Another special case is γ = 0, when the equation reads The motivation to study parabolic equations involving the normalized p-Laplacian stems partially from connections to time-dependent tug-of-war games [29,31,20] and image processing [18]. For regularity results concerning this equation, we refer to [6,3,8,21,19].
In this paper we continue the study of C 1,α -regularity by focusing on the range γ ∈ (−1, 0). Our main result is the following. Theorem 1.1. Let u be a viscosity solution of equation (1.1), where −1 < γ < 0, 1 < p < ∞, and f is a continuous and bounded function. There exist α = α(p, n, γ) ∈ (0, 1) and C = C(p, n, γ, ||u|| L ∞ (Q 1 ) , ||f || L ∞ (Q 1 ) ) > 0 such that when (x, t), (y, where β := α 2−αγ and σ := 1+α 2−αγ . Our proof relies on the method of alternatives and the improvement of flatness. The strategy is to define a process that provides a better linear approximation in a smaller cylinder, and which we can iterate until we reach a cylinder where a so called smooth alternative holds. More precisely, we define an induction hypothesis based on the size of the slope, see Corollary 3.3. In order to proceed with the iteration, we use an intrinsic scaling together with the approximation lemma [2,Lemma 4.1]. This lemma enables us to consider the solution of (1.1) as a perturbation of the solution of the corresponding homogeneous equation when the absolute size of f is sufficiently small. We may assume this by scaling.
The induction step can only work indefinitely if the gradient vanishes. In the case the iteration stops, our strategy is to show that the solution is sufficiently close to a linear function with a non-vanishing gradient. We show that in that case the solution itself has a gradient that is bounded away from zero in some cylinder. Hence, the equation is uniformly parabolic and no longer singular, so we can use the general regularity result from [27,28].
We remark that this method is flexible enough to be applied to the study of the gradient regularity for solutions of a more general class of singular, fully nonlinear parabolic equations of the type u t − |Du| γ F (D 2 u) = f, or those considered in [15], once the regularity for the corresponding homogeneous case has been treated.
This paper is organised as follows. In Section 2 we fix the notation and provide a Lipschitz estimate (Lemma 2.3), which is used in the non-singular alternative. In Section 3 we prove lemmas related to the improvement of flatness and iteration, and in Section 4 we prove Theorem 1.
where B r (x) ⊂ R n is a ball centered at x with radius r > 0. We will also use intrinsic parabolic cylinders We define viscosity (super-, sub-) solutions of equation (1.1) as follows. (1) For any φ ∈ C 2,1 (Q 1 ), Dφ(x 0 , t 0 ) = 0, touching u from below at (x 0 , t 0 ), it holds Similarly, viscosity subsolutions are defined changing touching from below by touching from above, inf by sup, and ≥ by ≤.
A continuous function is a viscosity solution of (1.1) if it is both viscosity sub-and supersolution.
Without a loss of generality, we assume that u(0, 0) = 0, ||u|| L ∞ (Q 1 ) ≤ 1 2 and ||f || L ∞ (Q 1 ) ≤ ε 0 , where ε 0 > 0 will be fixed in Lemma 3.1 below. Indeed, we can use the scaling Intermediate lemmas. In this section we gather some intermediate lemmas that will play a role in the proof of the Hölder regularity of the spatial gradient. First we recall the following result on the Lipschitz estimates for solutions to (1.1).
. Let u be a bounded viscosity solution to equation (1.1). There exists a constant C = C(p, n, γ) > 0 such that for all (x, t), (y, t) ∈ Q 7/8 , it holds Next, we consider bounded solutions of ≤ 1 and |K| ≥ 1. The previous Lemma provides a first Lipschitz estimate for w. Indeed, since w(x, t) + K · x is a solution to (1.1), 1 ≤ |K| and ||w|| L ∞ (Q 1 ) ≤ 1, we get for someC =C(p, n, γ). However, we can improve this estimate and provide a better control on the gradient. This estimate will play a key role in the non-singular alternative. Lemma 2.3. Let −1 < γ < 0 and 1 < p < ∞. Let K ∈ R n with 2 ≤ |K| ≤ M for some M = M (p, n, γ) > 0. Let w be a viscosity solution of (2.1), with w(0, 0) = 0. There exists The proof makes use of the Ishii-Lions method. It proceeds in two steps: first we obtain good enough Hölder estimates, and then use the Hölder estimate and the Ishii-Lions method again to prove the desired Lipschitz estimates. We postpone the technical proof to Section 5 in order to keep the paper easier to read.

Approximation Lemmas and iteration
In this section we state the approximation lemmas needed to implement the iteration, and define the induction hypothesis. Let us first recall an approximation result from [2].

then there exists a solutionũ to
Next we define a first linear approximation lemma by combining the previous lemma with the regularity result of [22, Theorem 1.1]. Since uniform Lipschitz estimates for deviations from planes don't seem to be available in the singular case, we refine [2, Lemma 4.2] by adding the parameter η 1 . This allows us to handle the case where the iteration stops by making u as close as needed to a linear function with a non-vanishing gradient. For the following lemma, recall that Proof. Letũ be the viscosity solution to It is important to notice that B depends only on p, n, γ. We choose µ 0 ∈ (0, 5/8) such that for some δ ∈ (0, 1/2). Thus there exist two constants ρ and δ depending on p, n, γ such that The choice of τ determines the smallness of f .
From now on we may assume that ε 0 < η 1 . Next we treat the situation of vanishing slope. Now with ρ, δ, ε 0 as in lemma 3.2 and η 1 as in lemma 2.3, we have the following iteration. Corollary 3.3. Let u be a viscosity solution to (1.1) such that osc Q 1 u ≤ 1. Let η 1 be the constant coming from Lemma 2.3, and B and ε 0 the constants coming from Lemma 3.2.
then there exists a vector l k+1 such that Proof. We set C 1 = B + 2. For j = 0 we take l 0 = 0, and the result follows from Lemma 3.2, since osc Q 1 u ≤ 1. Suppose that the result of the Lemma 3.2 holds for j = 0, . . . , k. We are going to prove it for j = k + 1. Define By assumption, we have osc Going back to u, we have Scaling back, we conclude osc (x,t)∈Q

Proof of the Hölder regularity of the spatial derivatives
We are now in a position to prove the Hölder continuity of Du at the origin and the improved Hölder regularity of u with respect to the time variable. Then there exist α = α(p, n, γ) ∈ (0, 1) and C = C(p, n, γ) > 0 such that where β := α 2−αγ and σ := 1+α 2−αγ .
Proof. Let ρ and δ be the constants coming from Lemma 3.2. Let k be the minimum nonnegative integer such that the condition (3.2) does not hold. We can conclude from Lemma 3.3 that for any vector ξ with |ξ| ≤ 2(1 − δ) k , it holds where τ := log(1−δ) log(ρ) and . Next we treat differently the following two cases.
First case: k = ∞. The regularity result holds with Indeed, for all k ∈ N, there exists l k ∈ R n with |l k | ≤ 2(1 − δ) k such that osc (y,t)∈Q We conclude the result by using the characterization of functions with Hölder continuous gradient, see also [28,3].

A better control on the Lipschitz estimates for deviation from planes
In this section, our aim is to provide a proof for Lemma 2.3. We start with a suitable control on the Hölder norm of w and apply again the Ishii-Lions's method in order to get good enough Lipschitz estimates.
then w is locally Hölder continuous in space and for (x, t), (y, t) ∈ Q 7/8 it holds , withC being the constant coming from (2.2).
Remark 5.2. If one could adapt the result of Wang [35] (see also the work of Savin [33] in the elliptic case) and prove that small perturbation of smooth solutions to some uniformly parabolic equation with a small enough continuous source term are locally C 1,α , then the proof will proceed without those Lipschitz estimates. This was done in [14] for equation where the operator is uniformly parabolic, and the source term is continuous. A possible generalization of [14] in case of singular or degenerate operators remains to be done.