On the Lipschitz character of orthotropic p-harmonic functions

Abstract

We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every $p\ge 2$ and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.

Appendix: Lipschitz regularity with a nonlinear lower order term

In this section, we consider the functional

$$\begin{aligned} \mathfrak {G}_\delta (u,\Omega ')=\sum _{i=1}^N \int _{\Omega '} \Big [g_i(u_{x_i}) + G(x,u)\Big ]\,dx ,\qquad \Omega '\Subset \Omega ,\ u\in W^{1,p}_\mathrm{loc}(\Omega '). \end{aligned}$$

The lower order term $f\,u$ of the functional $\mathfrak {F}_\delta$ is thus replaced by a more general term $G(x,u)$. We assume that $G$ is a Carathéodory function and that for almost every $x\in \Omega$, the map

$$\begin{aligned} \xi \mapsto G(x,\xi ) \qquad \text {is}\ C^{1}\ \text {and convex.} \end{aligned}$$

We denote $f(x,\xi ):=G_\xi (x,\xi )$ and we assume that $f\in W^{1,h}_\mathrm{loc}(\Omega \times \mathbb {R})$, for some $h>N/2$. Finally, we assume that $G(x,\xi )$ satisfies the inequality

$$\begin{aligned} |G(x,\xi )|\le b(x)\,|u|^{\gamma } +a(x) \end{aligned}$$

(A.1)

where $1<p\le \gamma <p^*$ and $a, b$ are two non-negative functions belonging respectively to $L^{s}_\mathrm{loc}(\Omega )$ and $L^{\sigma }_\mathrm{loc}(\Omega )$ with $s>N/p$ and $\sigma >p^*/(p^*-\gamma )$.

Under assumption (A.1), all the local minimizers of $\mathfrak {G}_\delta$ are locally bounded, see [11, Theorem 7.5] and moreover, for every such minimizer $u$, for every $B_{r_0}\Subset B_{R_0} \Subset \Omega$,

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(B_{r_0})} \le M, \end{aligned}$$

where $M$ depends on $\Vert u\Vert _{W^{1,p}(B_{R_0})}, r_0, R_0, \Vert b\Vert _{L^{\sigma }(R_0)}$, and $\Vert a\Vert _{L^{s}(B_{R_0})}$.

Then we have:

Theorem A.1

Let $p\ge 2$ and let ${U}\in W^{1,p}_\mathrm{loc}(\Omega )$ be a local minimizer of the functional $\mathfrak {G}_\delta$. Then U is locally Lipschitz in $\Omega$.

Proof

We only explain the main differences with respect to the proof of Theorem 1.1. Since $G$ is convex with respect to the second variable, the functional $\mathfrak {G}$ is still convex. This implies that Lemma 2.3 remains true with the same proof. We then introduce the approximation of $G$:

$$\begin{aligned} G_{\varepsilon }(x,\xi ) = \int _{\mathbb {R}^N\times \mathbb {R}}G(x-y, \xi -\zeta )\,\rho _{\varepsilon }(y) \,\widetilde{\rho }_{\varepsilon }(\zeta )\,dy\,d\zeta , \end{aligned}$$

where $\rho _{\varepsilon }$ is the same regularization kernel as before, while $\widetilde{\rho }_{\varepsilon }$ is a regularization kernel on $\mathbb {R}$.

Given a local minimizer $U\in W^{1,p}_\mathrm{loc}(\Omega )$ and a ball $B\subset 2\,B\Subset \Omega$, there exists a unique $C^{2}$ solution $u_{\varepsilon }$ to the regularized problem

$$\begin{aligned} \min \left\{ \mathfrak {G}_\varepsilon (v;B)\, :\, v-U_\varepsilon \in W^{1,p}_0(B)\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathfrak {G}_\varepsilon (v;B)=\sum _{i=1}^N \int _B g_{i,\varepsilon }(v_{x_i})\, dx+\int _B G_{\varepsilon }(x,v)\, dx \end{aligned}$$

and $U_\varepsilon =U*\rho _{\varepsilon }$. Moreover, by [11, Remark 7.6] we have $u_\varepsilon \in L^\infty (B)$, with a bound on the $L^\infty$ norm uniform in $\varepsilon >0$. In order to simplify the notation, we simply write as usual $u$ and $f$ instead of $u_{\varepsilon }$ and $f_\varepsilon$. The Euler equation is now

$$\begin{aligned} \sum _{i=1}^N \int g'_{i,\varepsilon }(u_{x_i})\, \varphi _{x_i}\, dx+\int f(x,u)\, \varphi \, dx=0,\qquad \varphi \in W^{1,p}_0(B). \end{aligned}$$

When we differentiate the Euler equation with respect to some direction $x_j$, we obtain

$$\begin{aligned} \sum _{i=1}^N \int g_{i,\varepsilon }''(u_{x_i})\, u_{x_i\,x_j}\, \psi _{x_i}\, dx{ +}\int \left( f_{x_j}(x,u)+f_{\xi }(x,u)\,u_{x_j}\right) \,\psi \,dx=0,\qquad \psi \in W^{1,p}_0(B). \end{aligned}$$

We can then repeat the proof of Proposition 5.1 with this additional term $f_{\xi }(x,u)u_{x_j}$ which leads to the following analogue of (5.12):

$$\begin{aligned} \begin{aligned} \left( \int _{B_t} \left( \mathcal {U}-\frac{1}{2} \right) _{+}^{\frac{2^*}{2}\,p} \mathcal {U}^{2^*\,q} \,dx\right) ^{\frac{2}{2^*}}&\le C\,\frac{q^5}{(s-t)^2} \int _{B_s} \mathcal {U}^{2\,q+p}\,dx \\&\quad + C\,\frac{q^5}{(s-t)^2} \,\varepsilon \, \int _{B_s} \mathcal {U}^{2\,q+2}\, dx \\&\quad +C\,q^5\, \Vert \nabla _x f\Vert _{L^h}\,\left( \int _{B_s} \mathcal {U}^{(2\,q+1)\,h'}\,dx\right) ^\frac{1}{h'}\\&\quad +C\,q^5\, \Vert f_\xi \Vert _{L^h}\, \left( \int _{B_s}\mathcal {U}^{(2\,q+2)\,h'}\,dx\right) ^{\frac{1}{h'}}. \end{aligned} \end{aligned}$$

Using again Hölder’s inequality for the first three terms, we obtain inequality (5.14) where $\Vert \nabla f\Vert$ now represents the full gradient of $f$ with respect to both $x$ and $\xi$. The rest of the proof is the same and leads to a uniform Lipschitz estimate, as desired. $\square$