BMO-regularity for a degenerate transmission problem

TL;DR

This paper investigates a complex degenerate transmission problem involving a quasilinear operator, establishing new regularity results such as local boundedness, gradient estimates in BMO spaces, and H"older continuity of solutions.

Contribution

It introduces novel regularity estimates for degenerate transmission problems, overcoming challenges posed by the absence of representation formulas and degeneracy.

Findings

Proved local boundedness of weak solutions.

Established gradient estimates in BMO spaces.

Demonstrated solutions are of class C^{0,Log-Lip} across the interface.

Abstract

We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. For bounded interface data, we prove the local boundedness of weak solutions and establish an estimate for their gradient in ${\rm BMO}-$spaces. The latter implies solutions are of class $C^{0,{\rm Log-Lip}}$ across the interface. Relaxing the assumptions on the data, we establish local H\"older continuity for the solutions.