$C^{1,\alpha}$ regularity for elliptic equations with the general nonstandard growth conditions

Abstract

We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega$. We prove the global $C^{1, \alpha}$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1, \alpha}$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

Keywords

• nonstandard growth
• $c^{1, \alpha}$ regularity
• hölder continuity
• bounded weak solution
• partial differential equations