Lipschitz bounds for nonuniformly elliptic integral functionals in the plane

Abstract

We study local regularity properties of local minimizer of scalar integral functionals with controlled $(p,q)$-growth in the two-dimensional plane. We establish Lipschitz continuity for local minimizer under the condition $1<p\le q<\infty$ with $q<3p$ which improve upon the classical results valid in the regime $q<2p$. Along the way, we establish an $L^{\infty }$-$L^2$-estimate for solutions of linear uniformly elliptic equations in the plane which is optimal with respect to the ellipticity contrast of the coefficients.

References in corpus (8)

• Maximal regularity for local minimizers of non-autonomous functionals
• Lipschitz bounds and nonautonomous integrals
• Regularity for double phase problems at nearly linear growth
• On the regularity of minimizers for scalar integral functionals with $(p,q)$-growth
• Regularity of minimizers of autonomous convex variational integrals
• On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals
• Regularity for multi-phase problems at nearly linear growth
• Local boundedness of minimizers under unbalanced Orlicz growth conditions

Cited by in corpus (1)

• Local and global $C^{1,\beta }$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type