Local regularity for quasi-linear parabolic equations in non-divergence form

Abstract

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.

Introduction

We are interested in the regularity of viscosity solutions of the following degenerate or singular parabolic equation in non-divergence form:

$${\partial }_{t}u-{\left|Du\right|}^{\gamma }\left[\Delta u+(p-2)〈D^{2}u\frac{Du}{\left|Du\right|},\frac{Du}{\left|Du\right|}〉\right]=f\text{in}{Q}_1,$$

where $-1<\gamma <\infty$, $1<p<\infty$ and $f$ is a continuous and bounded function. Existence and uniqueness of solutions to (1.1) were proved in [13], where more general singular or degenerate parabolic equations were considered (see also [7], [33] and the references therein). In [13], Demengel established global Hölder regularity results for the solutions of the Cauchy–Dirichlet problem associated to (1.1), under the assumptions that $f$ is continuous and bounded in space and Hölder in time, and that the boundary data is Hölderian in space and Lipschitz in time.

In this work, we investigate the higher regularity of the solution $u$ to (1.1). We focus on interior regularity for the gradient, away from boundaries. Let us mention two special cases. The case $\gamma =0$ corresponds to the normalized $p$-Laplacian

$${\Delta }_p^{N}u≔\Delta u+(p-2)〈D^{2}u\frac{Du}{\left|Du\right|},\frac{Du}{\left|Du\right|}〉,$$

and the regularity of the gradient was studied in [3], [19], [23] using viscosity theory methods. The case $\gamma =p-2$ corresponds to the usual parabolic $p$-Laplace equations, and it was shown in [24] that bounded weak solutions and viscosity solutions are equivalent. From this equivalence, we get the Hölder regularity of the gradient for bounded $f$ using variational methods [14], [15], [29], [38]. Let us also mention that recently Parviainen and Vázquez [34] established an equivalence between the radial solutions of (1.1) and the radial solutions of the standard parabolic $\gamma +2$-Laplace equation posed in a fictitious dimension. Notice that the regularity theory for (1.1) does not fall into the classical framework of fully nonlinear uniformly parabolic equations studied in [36], [37] due to the lack of uniform ellipticity and the presence of singularities.

In this paper, we provide local Lipschitz estimates for solutions to (1.1) in the whole range $-1<\gamma <\infty$. For $\gamma >0$, we prove the Hölder regularity of the gradient. Recently, for $\gamma \ne 0$, the homogeneous case $f=0$ was treated by Imbert, Jin and Silvestre [20]. The case where $f$ depends only on $t$ can be handled using the results of [20], since $\tilde{u}(x,t):=u(x,t)-{\int }_0^{t}f\left(s\right)ds$ solves the homogeneous equation. If we assume more regularity on $f$, let us say $f\in C_{x,t}^{1,0}\left(Q_1\right)$, then one could adapt the argument of [20] by regularizing the equation and differentiating it, and then prove the Hölder continuity of the gradient of the solutions of (1.1) with a norm which will then depend on ${\left|\left|Df\right|\right|}_{L^{\infty }\left(Q_1\right)}$. Our study relies on a nonlinear method based on compactness arguments where we avoid differentiating the equation and assume only the continuity of $f$. There are different characterizations of pointwise $C^{1+\alpha ,\frac{1+\alpha }{2}}$ functions, and we will use the one relying on the rate of approximations by planes. The study is based on estimates which prove that the solution gets flatter and flatter, when zooming into the smaller scales. There are three key points: an improvement of flatness estimate, the method of alternatives and the intrinsic scaling technique. In the degenerate case $\gamma >0$, in order to prove the Hölder regularity of the gradient, one has to choose a suitable scaling that takes into account the structure of the equation. Indeed, when the equation degenerates, the solutions locally generate their own scaling (“intrinsic scaling”) according to the values of the diffusion coefficients. The main idea behind the intrinsic scaling technique is to study the equation not on all parabolic cylinders, but rather on those whose ratio between space and time lengths depend on the size of the solution itself on the same cylinder, according to the regularity considered [14], [35]. Specifically, in order to prove Hölder regularity of the gradient, we consider the so called intrinsic cylinders defined by

$$Q_r^{\lambda }(x_0,t_0)≔B(x_0,r)\times (t_0-{\lambda }^{-\gamma }{r}^2,t_0],$$

where the parameter $\lambda >0$ behaves like $\underset{Q_r^{\lambda }}{sup}\left|Du\right|\approx \lambda$ (see Sections 4 First alternative and improvement of flatness, 5 Handling the two alternatives and proof of the main theorem).

Our strategy is to combine an improvement of flatness method with the method of alternatives (the Degenerate Alternative and the Smooth Alternative). This procedure defines an iteration that stops in the case where we reach a cylinder where the Smooth Alternative holds. More precisely, using an iteration process and compactness arguments, our aim is to prove that there exist $\rho =\rho (p,n,\gamma )>0$ and $\delta =\delta (p,n,\gamma )\in (0,1)$ with $\rho <{(1-\delta )}^{\gamma +1}$ such that one of the two following alternative holds:

• Degenerate Alternative: For every $k\in \mathbb{N}$ there exists a vector $l_k$ with $\left|l_k\right|\le C{(1-\delta )}^k$ such that

  $$\underset{(x,t)\in Q_{r_k}^{{\lambda }_k}}{osc}(u(x,t)-l_k\cdot x)\le r_k{\lambda }_k,$$

  where $r_k≔{\rho }^k$, ${\lambda }_k≔{(1-\delta )}^k$ and $Q_{r_k}^{{\lambda }_k}≔B_{r_k}\left(0\right)\times (-r_k^2{\lambda }_k^{-\gamma },0]$. That is, we have an improvement of flatness at all scales.

• Smooth Alternative: The previous iteration stops at some step $k_0$, that is, $\left|l_{k_0}\right|\ge C{(1-\delta )}^{k_0}$, and we can show that the gradient of $u$ stays away from 0 in some cylinder and then use the known results for uniformly parabolic equations with smooth coefficients [30], [31].

Notice that the intrinsic scaling plays a role in the choice of the cylinders $Q_{r_k}^{{\lambda }_k}$ in order to proceed with the iteration, and that if $\left|l_k\right|\le C{(1-\delta )}^k$ for all $k$, then $\left|Du(0,0)\right|=0$. Let us explain how these alternatives appear. The existence of the vector $l_{k+1}$ in the iteration process can be reduced to the proof of an improvement of flatness (see Section 4) for the function

$$w_k(x,t)=\frac{u(r_{k}x,r_k^2{\lambda }_k^{-\gamma }t)-l_k\cdot r_{k}x}{r_k{\lambda }_k}.$$

The function $w_k$ solves

$${\partial }_t{w}_k-{|D{w}_k+\frac{l_k}{{\lambda }_k}|}^{\gamma }\left[\Delta w_k+(p-2)〈D^2{w}_k\frac{D{w}_k+l_k∕{\lambda }_k}{|D{w}_k+l_k∕{\lambda }_k|},\frac{D{w}_k+l_k∕{\lambda }_k}{|D{w}_k+l_k∕{\lambda }_k|}〉\right]=\bar{f}\text{in}{Q}_1,$$

where $\bar{f}(x,t)≔r_k{\lambda }_k^{-(\gamma +1)}f(r_{k}x,r_k^2{\lambda }_k^{-\gamma }t)$. This leads us to study the equation satisfied by the deviations of $u$ from planes $w(x,t)=u(x,t)-q\cdot x$,

$${\partial }_{t}w-{|Dw+q|}^{\gamma }\left[\Delta w+(p-2)〈D^{2}w\frac{Dw+q}{|Dw+q|},\frac{Dw+q}{|Dw+q|}〉\right]=\bar{f}\text{in}{Q}_1.$$

We see that $w_k$ satisfies (1.2) with $q=l_k∕{\lambda }_k$. The proof of the improvement of flatness is based on compactness estimates and a contradiction argument. Unlike the case of the normalized $p$-Laplacian, the ellipticity coefficients of Eq. (1.2) depend on $q$. To tackle this problem, we have to introduce the two alternatives: either we have a uniform bound on $\left|q\right|$ and we can run again our iteration, or $\left|q\right|$ is larger than some fixed constant. In this later case, using Lipschitz estimates in the space variable which are independent of $q$ (see Lemma 3.3), we can provide a strictly positive lower bound for the gradient of $u$ and finish the proof by using known results for uniformly parabolic equations with smooth coefficients. Our main result is the following.

Theorem 1.1

Let $0\le \gamma <\infty$ and $1<p<\infty$. Assume that $f$ is a continuous and bounded function, and let $u$ be a bounded viscosity solution of (1.1). Then $u$ has a locally Hölder continuous gradient, and there exist a constant $\alpha =\alpha (p,n,\gamma )$ with $\alpha \in (0,\frac{1}{1+\gamma })$ and a constant $C=C(p,n,\gamma )>0$ such that

$$|Du(x,t)-Du(y,s)|\le C\left(1+{\left|\left|u\right|\right|}_{L^{\infty }\left(Q_1\right)}+{\left|\left|f\right|\right|}_{L^{\infty }\left(Q_1\right)}\right)({|x-y|}^{\alpha }+{|t-s|}^{\frac{\alpha }{2}})$$

and

$$|u(x,t)-u(x,s)|\le C\left(1+{\left|\left|u\right|\right|}_{L^{\infty }\left(Q_1\right)}+{\left|\left|f\right|\right|}_{L^{\infty }\left(Q_1\right)}\right){|s-t|}^{\frac{1+\alpha }{2}}.$$

The method of the proof can apply to more general degenerate and fully nonlinear equations of the form $u_t-{\left|Du\right|}^{\gamma }F\left(D^{2}u\right)=f$, where $F$ is uniformly parabolic, once we show the regularity for the associated homogeneous equation. In the singular case $-1<\gamma <0$, we were not able to provide uniform (with respect to $q$) Lipschitz estimates for solutions to (1.2). The higher regularity of the gradient is still an open problem when $\gamma <0$.

The paper is organized as follows. In Section 2 we fix the notations, gather some known regularity results that we will use later on, and reduce the problem by re-scaling. Section 3 is devoted to the study of Eq. (1.2) and provides the needed compactness estimates. In Section 4, we provide the proof of the “improvement of flatness” property. In Section 5 we prove Theorem 1.1 proceeding by iteration and considering the two possible alternatives. Section 6 contains the proof of the Lipschitz regularity for solutions to (1.1) for $-1<\gamma <\infty$. In Section 7 we prove the uniform Lipschitz estimates for solutions to (1.2) for $0\le \gamma <\infty$.