Calculus of Variations and Geometric Measure Theory

S. Biagi - G. Cupini - E. Mascolo

Regularity of quasi-minimizers for non-uniformly elliptic integrals

created by cupini on 24 Jun 2019


Submitted Paper

Inserted: 24 jun 2019
Last Updated: 24 jun 2019

Year: 2019


In this paper we consider a class of non-uniformly elliptic integral functionals $\mathcal{F}$ and we prove the local boundedness of the quasi-minimizers of $\mathcal{F}$. As regards the integrand function $f$ defining $\mathcal{F}$, we require that $$\lambda(x)\,
p\leq f(x,u,\xi)\leq \mu(x)\,(
q)+a(x), $$ where $\lambda,\mu,a$ are measurable functions satisfying suitable integrability assumpions. Our approach is based on a suitable adaptation of the celebrated De Giorgi proof, and it relies on an appropriate Caccioppoli-type inequality.

Keywords: local boundedness, Non-uniformly elliptic functionals, regularity of quasi-minimizers, Caccioppoli-type inequality