Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure

TL;DR

This paper develops a new regularity theory for non-autonomous PDEs without relying on traditional structure conditions, broadening the scope of ellipticity and continuity assumptions for solutions.

Contribution

It introduces a novel ellipticity condition and relaxes structure assumptions, enabling regularity results without Uhlenbeck structure or special function dependencies.

Findings

Established local $C^{1,eta}$ regularity for solutions.

Extended regularity results to broader classes of non-autonomous PDEs.

Included known optimal regularity results as special cases.

Abstract

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_\Omega F(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or $F$ with general $(p,q)$-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity $F$ satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as $t^p$, $\varphi(t)$, $t^{p(x)}$, $t^p+a(x)t^q$, and not only $F$ but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using $A$ or $F$ only, which entails that the function is quasi-isotropic, i.e.\ it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on $A$ or $F$ without specific structure and without direct restriction on the ratio $\frac qp$ of the parameters from the $(p,q)$-growth condition. We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.