The two-phase Alt-Phillips problem for quasilinear operators

TL;DR

This paper proves interior regularity, growth estimates, and boundary properties for minimizers of a quasilinear Alt-Phillips functional across all p-values and nonlinearity powers, advancing understanding of free boundary problems.

Contribution

It establishes new regularity, growth, and geometric measure results for sign-changing minimizers of the quasilinear Alt-Phillips problem for all relevant parameters.

Findings

Proved interior regularity and optimal growth estimates for minimizers.

Established local finite perimeter and density estimates for free boundaries.

Showed rectifiability and Hausdorff measure finiteness of free boundaries.

Abstract

We establish interior regularity and optimal growth estimates for sign-changing minimizers of the $p-$singular or $p-$degenerate quasilinear Alt--Phillips functional throughout the full range of $1<p<\infty$ and of the nonlinearity power $0<\gamma<p$. In addition, we obtain local finite perimeter and density estimates, from which we deduce the local $(N-1)$-rectifiability of the reduced and two-phase free boundaries and the local finiteness of their $(N-1)$-dimensional Hausdorff measure for a restricted range of $\gamma$.