Local boundedness for solutions of a class of non-uniformly elliptic anisotropic problems

TL;DR

This paper establishes local boundedness of solutions to a class of nonlinear, non-uniformly elliptic anisotropic problems with complex growth conditions, advancing understanding of regularity in such equations.

Contribution

It proves local boundedness for scalar local quasi-minimizers of energy integrals with anisotropic p_i,q-growth conditions, extending regularity results to more general elliptic problems.

Findings

Solutions are locally bounded under specified growth conditions.

The results apply to equations with anisotropic and non-uniform ellipticity.

The work broadens the class of problems where regularity can be assured.

Abstract

We consider a class of {energy integrals}, associated to nonlinear and non-uniformly elliptic equations, with integrands $f(x,u,\xi)$ satisfying anisotropic $p_i,q$-growth conditions of the form $$ \sum_{i=1}^n \lambda_i (x)|\xi_i|^{p_i}\le {f}(x,u,\xi)\le \mu (x)\left\{|\xi|^{q} + |u|^{\gamma}+1\right\} $$ for some exponents $\gamma\ge q\geq p_i>1$, and non-negative functions $\lambda_i,\mu$ subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.