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

Abstract

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_ΩF(x,Du)\,dx, \] 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 $t^p$, $\phi (t)$, $t^{p(x)}$, $t^p+a(x)t^q$, 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 $\frac{q}{p}$ 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.

Arch. Ration. Mech. Anal., to appear

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