A transmission problem for $(p,q)$-Laplacian

TL;DR

This paper studies a double-phase variational problem involving the $(p,q)$-Laplacian, establishing existence, regularity of minimizers, and smoothness of the free boundary across the zero level surface.

Contribution

It proves the existence and Hölder continuity of minimizers, derives a weak free boundary condition, and shows the free boundary is $C^{1,eta}$ almost everywhere.

Findings

Existence of minimizers for the double-phase functional.

Hölder continuity of the minimizers.

$C^{1,eta}$ regularity of the free boundary almost everywhere.

Abstract

In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level "surface" of the minimiser of the functional $$ J(v,\Omega) = \int_\Omega \left( |D v^+|^p + |D v^-|^q \right) dx. $$ We prove that a minimiser exists, and is H\"older continuous, whence using an intrinsic variation we prove a weak formulation of the free boundary condition across the zero level surface, formally represented by $$ (q-1)|D u^-|^q = (p-1) |D u^+|^p, \quad \hbox{on } \partial \{u > 0\}. $$ We show that the free boundary is $C^{1,\alpha}$ a.e. with respect to the measure $\Delta_p u^+$, whose support is of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure.