Hölder gradient estimates for a class of singular or degenerate parabolic equations

We prove interior H ¨ older estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation


Introduction
Let 1 < p < ∞ and κ ∈ (1 − p, ∞).We are interested in the regularity of solutions of u t = |∇u| κ div(|∇u| p−2 ∇u). ( When κ = 0, this is the classical parabolic p-Laplacian equation in divergence form.This is the natural case in the context of gradient flows of Sobolev norms.Hölder estimates for the spatial gradient of their weak solutions (in the sense of distribution) were obtained by DiBenedetto and Friedman in [7] (see also Wiegner [24]).
The equation (1) can be rewritten as where γ = p + κ − 2 > −1.In this paper, we prove Hölder estimates for the spatial gradients of viscosity solutions to (2) for 1 < p < ∞ and γ ∈ (−1, ∞).Therefore, it provides a unified approach for all those γ and p, including the two special cases γ = 0 and γ = p − 2 mentioned above.
Our proof in this paper follows a similar structure as in [12], with some notable differences that we explain below.We use non-divergence techniques in the context of viscosity solutions.Theorem 1.1 tells us that these techniques are in some sense stronger than variational methods when dealing with the regularity of scalar p-Laplacian type equations.The weakness of these methods (at least as of now) is that they are ineffective for systems.
The greatest difficulty extending the result in [12] to Theorem 1.1 comes from the lack of uniform ellipticity.When γ = 0, the equation ( 2) is a parabolic equation in non-divergence form with uniformly elliptic coefficients (depending on the solution u).Because of this, in [12], we use the theory developed by Krylov and Safonov, and other classical results, to get some basic uniform a priori estimates.This fact is no longer true for other values of γ.The first step in our proof is to obtain a Lipschitz modulus of continuity.That step uses the uniform ellipticity very strongly in [12].In this paper we take a different approach using the method of Ishii and Lions [10].Another step where the uniform ellipticity plays a strong role is in a lemma which transfers an oscillation bound in space, for every fixed time, to a space-time oscillation.In this paper that is achieved through Lemmas 4.4 and 4.5, which are considerably more difficult than their counterpart in [12].Other, more minor, difficulties include the fact that the non-homogeneous right hand side forces us to work with a different scaling (See the definition of Q ρ r by the beginning of Section 4).In order to avoid some of the technical difficulties caused by the non-differentiability of viscosity solutions, we first consider the regularized problem (3) in the below, and then obtain uniform estimates so that we can pass to the limit in the end.For ε ∈ (0, 1), let u be smooth and satisfy that We are going to establish Lipschitz estimate and Hölder gradient estimates for u, which will be independent of ε ∈ (0, 1), in Sections 2, 3, 4. Then in Section 5, we recall the definition of viscosity solutions to (2), as well as their several useful properties, and prove Theorem 1.1 via approximation arguments.This idea of approximating the problem with a smoother one and proving uniform estimates is very standard.
Acknowledgement: Part of this work was done when T. Jin was visiting California Institute of Technology as an Orr foundation Caltech-HKUST Visiting Scholar.He would like to thank Professor Thomas Y. Hou for the kind hosting and discussions.

Lipschitz estimates in the spatial variables
The proof of Lipschitz estimate in [12] for γ = 0 is based on a calculation that |∇u| p is a subsolution of a uniformly parabolic equation.We are not able to find a similar quantity for other nonzero γ.The proof we give here is completely different.It makes use of the Ishii-Lions' method [10].However, we need to apply this method twice: first we obtain log-Lipschitz estimates, and then use this log-Lipschitz estimate and Ishii-Lions' method again to prove Lipschitz estimate.Moreover, the Lipschitz estimate holds for γ > −2 instead of γ > −1.
Proof.Without loss of generality, we assume x 0 = 0 and t 0 = 0.It is sufficient to prove that is non-positive, where We assume this is not true and we will exhibit a contradiction.In the rest of the proof, t ∈ [−1, 0] and x, y ∈ B 1 denote the points realizing the maximum defining M .
Since M ≥ 0, we have In particular, , where δ = |a| and a = x − y, and Hence, for L 2 large enough, depending only on u L ∞ (Q 4 ) , we can ensure that t ∈ (−1, 0] and x, y ∈ B 1 .We choose L 2 here and fix it for the rest of the proof.Thus, from now on L 2 is a constant depending only on u L ∞ .Choosing L 1 large, we can ensure that δ(< e −2 ) is small enough to satisfy φ(δ) ≥ 2δ.
In this case, (4) implies Since t ∈ [−1, 0] and x, y ∈ B 1 realizing the supremum defining M , we have that where and q = L 1 φ ′ (δ)â, X = ∇ 2 u(t, x) and Y = ∇ 2 u(t, y).By evaluating the equation at (t, x) and (t, y), we have Whenever we write C in this proof, we denote a positive constant, large enough depending only on n, p, γ and u L ∞ (Q 4 ) , which may vary from lines to lines.Recall that we have already chosen L 2 above depending on u L ∞ only.
Note that |q| = L 1 |φ ′ (δ)|.Choosing L 1 large enough, δ will be small, |φ ′ (δ)| will thus be large, and |q| ≫ L 2 .In particular, From (7) and the fact that φ ′′ (δ) < 0, we have Making use of ( 8), ( 9) and (10), we have Therefore, it follows from (10) and the ellipticity of A that Similarly, We get from ( 8) and ( 5) the following inequality where We first estimate T 2 .Using successively (5), ( 9), (11) and mean value theorem, we get We now turn to T 1 .On one hand, evaluating (7) with respect to a vector of the form (ξ, ξ), we get that for all ξ ∈ R d we have On the other hand, when we evaluate (7) with respect to (â, â), we get, The inequality (14) tells us that all eigenvalues of (X − Y ) are bounded above by a constant C. The inequality (15) tells us that there is at least one eigenvalue that is less than the negative number 4L 1 φ ′′ (δ) + 2L 2 .Because of the uniform ellipticity of A, we obtain In view of the estimates for T 1 and T 2 , we finally get from (12) that or equivalently, Our purpose is to choose L 1 large in order to get a contradiction in (16).
For L 1 sufficiently large, since γ > −2 The remaining term is handled because of the special form of the function φ.We have for L 1 sufficiently large.Therefore, we reached a contradiction.The proof of this lemma is thereby completed.
By letting t = t 0 and y = x 0 in Lemma 2.1, and since (x 0 , t 0 ) is arbitrary, we have Corollary 2.2.Let u be a smooth solution of (3) in Q 4 with γ > −2 and ε ∈ (0, 1).Then there exists a positive constant C depending only on n, γ, p and u L ∞ (Q 4 ) such that for every (t, x), (t, y) ∈ Q 3 and |x − y| < 1/2, we have We shall make use of the above log-Lipschitz estimate and the Ishii-Lions' method [10] again to prove the following Lipschitz estimate.
Proof.The proof of this lemma follows the same computations as that of Lemma 2.1, but we make use of the conclusion of Corollary 2.2 in order to improve our estimate.Without loss of generality, we assume x 0 = 0 and t 0 = 0.As before, we define is non-positive, where We assume this is not true in order to obtain a contradiction.In the remaining of the proof of the lemma, t ∈ [−1, 0] and x, y ∈ B 1/4 denote the points realizing the maximum defining M .
For the same reasons as in the proof of Lemma 2.1, the inequalities ( 4) and ( 5) also apply in this case.Thus, we can use the same choice of L 2 , depending on u L ∞ only, that ensures t ∈ (−1, 0] and x, y ∈ B 1 . From Corollary 2.2, we already know that u(t, x) − u(t, y) In particular, we obtain an improvement of ( 5), This gives us an upper bound for |x + y| that we can use to improve (13).
The estimate for T 1 stays unchanged.Hence, (16) becomes The term +1 inside the innermost parenthesis is there just to ensure that the inequality holds both for γ < 0 and γ > 0. Recalling that δ < C/L 1 , we obtain an inequality in terms of L 1 only.
Choosing L 1 large, we arrive to a contradiction given that Again, by letting t = t 0 and y = x 0 in Lemma 2.3, and since (x 0 , t 0 ) is arbitrary, we have Corollary 2.4.Let u be a smooth solution of (3) in Q 4 with γ > −2 and ε ∈ (0, 1).Then there exists a positive constant C depending only on n, γ, p and u L ∞ (Q 4 ) such that for every (t, x), (t, y) ∈ Q 3 and |x − y| < 1, we have

Hölder estimates in the time variable
Using the Lipschitz continuity in x and a simple comparison argument, we show that the solution of ( 3) is Hölder continuous in t.
Lemma 3.1.Let u be a smooth solution of (3) in Q 4 with γ > −1 and ε ∈ (0, 1).Then there holds sup where C is a positive constant depending only on n, p, γ and u L ∞ (Q 4 ) .

Remark 3.2. Deriving estimates in the time variable for estimates in the space variable by maximum principle techniques is classical.
As far as viscosity solutions are concerned, the reader is referred to [4,Lemma 9.1,p. 317] for instance.
Proof.Let β = max(2, (2 + γ)/(1 + γ)).We claim that for all t 0 ∈ [−1, 0), η > 0, there exists We first choose (19) holds true for x ∈ ∂B 1 .We will next choose L 2 such that (19) holds true for t = t 0 .In this step we shall use Corollary 2.4 that u is Lipschitz continuous with respect to the spatial variables.From Corollary 2.4, We finally choose L 1 such that the function ϕ(t, x) is a supersolution of an equation that u is a solution.The inequality (19) thus follows from the comparison principle.We use a slightly different equation depending on whether γ ≤ 0 or γ > 0.
Let us start with the case γ ≤ 0. In this case we will prove that ϕ is a supersolution of the nonlinear equation (3).That is In order to ensure this inequality, we choose L 1 so that We chose the exponent β so that when γ ≤ 0, |∇ϕ| γ |D 2 ϕ| = CL 1+γ 1 for some constant C depending on n and γ.Thus, we must choose L 1 = CL 1+γ 2 in order to ensure (20).Therefore, still for the case γ ≤ 0, β = (2 + γ)/(1 + γ), and for any choice of η > 0, using the comparison principle, The lemma is then concluded in the case γ ≤ 0.
Let us now analyze the case γ > 0. In this case, we prove that ϕ is a supersolution to a linear parabolic equation whose coefficients depend on u.That is Since γ > 0 and ∇u is known to be bounded after Corollary 2.4, we can rewrite the equation assumption where the coefficients a ij (t, x) are bounded by Since γ > 0, we pick β = 2 and D 2 ϕ is a constant multiple of L 2 .In particular, we ensure that (22) holds if Therefore, for the case γ > 0, β = 2, and for any choice of η > 0, using the comparison principle, This finishes the proof for γ > 0 as well.

Hölder estimates for the spatial gradients
In this section, we assume that γ > −1 so that Corollary 2.4 and Lemma 3.1 holds, that is, the solution of (3) in Q 2 has uniform interior Lipschitz estimates in x and uniform interior Hölder estimates in t, both of which are independent of ε ∈ (0, 1).For ρ, r > 0, we denote The cylinders Q ρ r are the natural ones that correspond to the two-parameter family of scaling of the equation.Indeed, if u solves (3) in Q ρ r and we let v(x, t) If we choose ρ ≥ ∇u L ∞ (Q 1 ) + 1, we may assume that the solution of (3 We are going to show that ∇u is Hölder continuous in space-time at the point (0, 0).The idea of the proof in this step is similar to that in [12].First we show that if the projection of ∇u onto the direction e ∈ S n−1 is away from 1 in a positive portion of Q 1 , then ∇u • e has improved oscillation in a smaller cylinder.Lemma 4.1.Let u be a smooth solution of (3) with ε ∈ (0, 1) such that |∇u| ≤ 1 in Q 1 .For every 1  2 < ℓ < 1, µ > 0, there exists τ 1 ∈ (0, 1  4 ) depending only on µ, n, and there exist τ, δ > 0 depending only on n, p, γ, µ and ℓ such that for arbitrary e ∈ S n−1 , if and denote Differentiating (3) in x k , we have we have Then in the region Ω + = {(x, t) ∈ Q 1 : w > 0}, we have Since |∇u| > ℓ/2 in Ω + , we have in Ω + : where c(p, n, γ) is a positive constant depending only on p, n and γ.By Cauchy-Schwarz inequality, it follows that where for some constant c 0 > 0 depending only on p, γ, n.Therefore, it satisfies in the viscosity sense that where Notice that since ℓ ∈ ( 1 2 , 1), ãij is uniformly elliptic with ellipticity constants depending only on p and γ.We can choose c 2 (ℓ) > 0 depending only on p, γ, n and ℓ such that if we let then we have )/c 2 .Therefore, it follows from the assumption that By Proposition 2.3 in [12], there exist τ 1 > 0 depending only µ and n, and ν > 0 depending only on µ, ℓ, n, γ and p such that Meanwhile, we have This implies that Therefore, we have Since |∇u • e| ≤ |∇u|, we have Therefore, remarking that ν ≤ 1 + ρ, we have for some δ > 0 depending only on p, γ, µ, ℓ, n.Finally, we can choose τ = τ 1 if γ < 0 and Note that our choice of τ and δ in the above implies that when γ ≥ 0.
In the rest of the paper, we will choose τ even smaller such that This fact will be used in the proof of Theorem 4.8.
In case we can apply the previous lemma holds in all directions e ∈ ∂B 1 , then it effectively implies a reduction in the oscillation of ∇u in a smaller parabolic cylinder.If such improvement of oscillation takes place at all scales, it leads to the Hölder continuity of ∇u at (0, 0) by iteration and scaling.The following corollary describes this favorable case in which the assumption of the previous Lemma holds in all directions.Corollary 4.2.Let u be a smooth solution of (3) with ε ∈ (0, 1) such that |∇u| ≤ 1 in Q 1 .For every 0 < ℓ < 1, µ > 0, there exist τ ∈ (0, 1/4) depending only on µ and n, and δ > 0 depending only on n, p, γ, µ, ℓ, such that for every nonnegative integer k for all i = 0, • • • , k.
Proof.When i = 0, it follows from Lemma 4.
Then v satisfies By the induction hypothesis, we also know that |∇v| ≤ 1 in Q 1 , and Notice that ε ≤ (1 − δ) k .Therefore, by Lemma 4.1 we have .
Unless ∇u(0, 0) = 0, the above iteration will inevitably stop at some step.There will be a first value of k where the assumptions of Corollary 4.2 do not hold in some direction e ∈ S n−1 .This means that ∇u is close to some fixed vector in a large portion of Q . We then prove that u is close to some linear function, from which the Hölder continuity of ∇u will follow applying a result from [23].
Having ∇u close to a vector e for most points tells us that for every fixed time t, the function u(x, t) will be approximately linear.However, it does not say anything about how u varies respect to time.We must use the equation in order to prove that the function u(x, t) will be close to some linear function uniformly in t.That is the main purpose of the following set of lemmas.
where C is a positive constant depending only on M, γ, p and the dimension n.
We claim that We only justify the first inequality since we can proceed similarly to get the second one.If not, let m = − inf Q 1 (w − u) > 0 and (x 0 , t 0 ) ∈ Q 1 be such that m = u(x 0 , t 0 ) − w(x 0 , t 0 ).Then w + m ≥ u in Q 1 and w(x 0 , t 0 ) + m = u(x 0 , t 0 ).By the choice of ā, we know that which is impossible.Therefore, x 0 ∈ B 1 .But this is not possible since w is a strict super-solution of the equation satisfied by u.This proves the claim.Therefore, we have . Let e ∈ S n−1 and 0 < δ < 1/8.Assume that for all t ∈ [−1, 0], we have where C is a positive constant depending only on γ, p and the dimension n.
Let η be the one in Theorem 4.7 with β = 1/2, and for this η, let ε 0 , ε 1 be two sufficiently small positive constants so that the conclusion of Lemma 4.6 holds.For from which it follows that Let τ, δ be the constants in Corollary 4.2.Denote [log ε/ log(1 − δ)] as the integer part of log ε/ log(1 − δ).Let k be either [log ε/ log(1 − δ)] or the minimum nonnegative integer such that the condition (26) does not hold, whichever is smaller.Then it follows from Corollary 4.2 that for all ℓ = 0, 1, where C = 1 1−δ and α = log(1−δ) log τ .Thus, for every q ∈ R n such that |q| ≤ (1 − δ) k .Note that when γ ≥ 0, it follows from (25) that Then and thus, osc Therefore, when ℓ = k, the equation ( 34) is a uniformly parabolic quasilinear equation with smooth and bounded coefficients.By the standard quasilinear parabolic equation theory (see, e.g., Theorem 4.4 of [15] in page 560) and Schauder estimates, there exists where C > 0 depends only on γ, p and n, and we used that α 2−αγ ≤ 1 2 .Rescaling back, we have and Then we can conclude from (31) and (36) that where C > 0 depends only on γ, p and n.From (37), we obtain that for where in the last inequality we have used (25).From ( 35) and (38), we have for all t ∈ (−1/4, 0], where β is chosen such that That is, Consequently, using (29), we get It follows from Lemma 4.6 that there exists a ∈ R such that By Theorem 4.7, there exists b ∈ R n such that Rescaling back, we have .
Together with (31) and (35), we can conclude as in Case 1 that In conclusion, we have proved that there exist q ∈ R n with |q| ≤ 1, and two positive constants α, C depending only on γ, p and n such that where β is given in (39).Then the conclusion follows from standard translation arguments.

Approximation
As mentioned in the introduction, the viscosity solutions to with γ > −1 and p > 1 fall into the general framework studied by Ohnuma-Sato in [19], which is an extension of the work of Barles-Georgelin [5] and Ishii-Souganidis [11] on the viscosity solutions of singular/degenerate parabolic equations.Let us recall the definition of viscosity solutions to (40) in [19].We denote Let F be the set of functions f ∈ C 2 ([0, ∞)) satisfying and This set F is not empty when γ > −1 and p > 1, since f (r) = r β ∈ F for any β > max( γ+2 γ+1 , 2).Moreover, if f ∈ F, then λf ∈ F for all λ > 0.

A Appendix
We will adapt some arguments in [6] to prove Theorem 5.5.In the following, c denotes some positive constant depending only on n, γ and p, which may vary from line to line.Denote Lemma A.1.For every z ∈ ∂B 1 , there exists a function W z ∈ C(B 1 ) such that W z (z) = 0, W z > 0 in B 1 \ {z}, and Then there exists δ > 0 depending only on n, γ and p such that for x ∈ B 1 ∩ B 1+δ (2z), we have where in the first inequality we used the choice of σ.Then we choose a that Since w z (z) = 0 and G z (z) > 0, the function agrees with w z in a neighborhood of z (relative to B 1 ).Also, because of the choice of a, W z agrees with G z when x ∈ B 1 and |x − 2z| ≥ 1 + δ for some δ ∈ (0, δ).Moreover, for some constant κ > 0 depending only on n, γ and p.By multiplying a large positive constant to W z , we finish the proof of this lemma.

Lemma A.2. For every
Proof.For τ > −1 and x ∈ ∂B 1 , then is a desired function, where W z is the one in Lemma A.1.For τ = −1 and x ∈ B 1 , we let where β = max( γ+2 γ+1 , 2).Then if we choose A > 0 large, which depends only on n, γ and p, then W z,τ will be a desired function.
For two real numbers a and b, we denote a ∨ b = max(a, b), a ∧ b = min(a, b).
be a solution of (3) with γ > −1 and ε ∈ (0, 1).Let ϕ := u| ∂pQ 1 and let ρ be a modulus of continuity of ϕ.Then there exists another modulus of continuity ρ * depending only on n, γ, p, ρ such that Proof.For every κ > 0 and (z, τ would suffice, and is independent of the choice of (z, τ ).Finally, let Note that for every κ > 0 and (z, τ where ω is the modulus of continuity for W z,τ , which is evidently independent of (z, τ ).Let ρ(r) = inf κ>0 (κ + M κ ω(r)) for all r ≥ 0. Then ρ is a modulus of continuity, and By Lemma A.2, W κ,z,τ is a supersolution of (3) for every κ > 0 and (z, τ ) ∈ ∂ p Q 1 , and therefore, W is also a supersolution of (3).By the comparison principle, This finishes the proof of this theorem.