An Extension of De Giorgi Class and Applications

TL;DR

This paper extends the De Giorgi class to establish local boundedness and H"older continuity of solutions, with applications to polyconvex functionals, degenerate elliptic equations, non-standard growth conditions, and quasilinear systems.

Contribution

It introduces a generalized De Giorgi class and demonstrates its utility in proving regularity for various complex elliptic problems.

Findings

Functions in the extended class are locally bounded and H"older continuous.

Regularity results apply to polyconvex functionals with complex energy densities.

Solutions to degenerate and non-standard elliptic equations are also shown to be regular.

Abstract

We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H\"older continuous. Some applications are given. As a first application, we give a regularity result for local minimizers $u:\Omega \subset \mathbb R^4 \rightarrow \mathbb R^4$ of a special class of polyconvex functionals with splitting form in four dimensional Euclidean spaces. Under some structural conditions on the energy density, we prove that each component $u^\alpha$ of the local minimizer $u$ belongs to the generalized De Giorgi class, then one can derive that it is locally bounded and locally H\"older continuous. Our result can be applied to polyconvex integrals whose prototype is $$ \int_\Omega \Big(\sum_{\alpha =1}^4 |Du^\alpha|^p + \sum_{\beta =1}^6 |({\rm adj}_2 Du )^\beta | ^q +\sum_{\gamma =1}^4 |({\rm adj}_3 Du )^\gamma | ^r +|\det Du|^s \Big ) \mathrm {d}x $$ with suitable $p,q,r,s\ge 1$. As a second application, we consider a degenerate linear elliptic equation of the form $$ -\mbox {div} (a(x)\nabla u)=-\mbox {div}F, $$ with $0<a(x) \le \beta <+\infty$. We prove, by virtue of the generalized De Giorgi class, that any weak solution is locally bounded and locally H\"older continuous provided that $\frac 1 {a(x)}$ and $F(x)$ belong to some suitable locally integrable function spaces. As a third application, we show that our theorem can be applied in dealing with regularity issues of elliptic equations with non-standard grow conditions. As a fourth application we treat with quasilinear elliptic systems. Under suitable assumptions on the coefficients, we show that any of its weak solutions is locally bounded and locally H\"older continuous.