partial differential equations

Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth

arXiv:2505.14089

summary

The paper studies elliptic obstacle problems with generalized Orlicz growth and measure data, proving existence of solutions in Musielak‑Orlicz spaces and obtaining gradient estimates via nonlinear Wolff potentials that lead to $C^{1, α}$ regularity.

Abstract

This paper investigates elliptic obstacle problems with generalized Orlicz growth involving measure data, which includes Orlicz growth, variable exponent growth, and double-phase growth as specific cases of this setting. First, we establish the existence of solutions in the Musielak-Orlicz space. Then, we derive pointwise and oscillation gradient estimates for solutions in terms of the non-linear Wolff potentials, assuming minimal conditions on the obstacle. These estimates subsequently lead to $C^{1, I^\pm}$ -regularity results for the solutions.

There is a technical gap in the proof of Lemma 4.5 concerning the boundary data

Topics & keywords

#elliptic obstacle problems #generalized orlicz growth #wolff potentials #regularity theory #measure dataMusielak-Orlicz spacenonlinear Wolff potentialC^{1,α} regularityvariable exponent growthdouble-phase growth