The two-phase Alt-Phillips problem for quasilinear operators

Abstract

We establish interior regularity and growth estimates for the 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 $0<\gamma <p$. As a consequence, we show that the nonbranching hypersurface is locally $C^{1,\alpha }$ rectifiable. In addition, we obtain local density estimates, from which we deduce the local finiteness of the $(N-1+\gamma (p-1)/(p-\gamma ))-$dimensional Hausdorff measure of the remaining free boundaries for a restricted range of $\gamma$.

A revised version; to appear in Interfaces Free Bound