The free boundary for a superlinear system

TL;DR

This paper investigates the regularity and analyticity of free boundaries arising in superlinear systems, using variational, viscosity, and linearization techniques to establish optimal regularity and smoothness properties.

Contribution

It introduces new regularity results for minimizers and free boundaries in superlinear systems, including $C^{1,eta}$ regularity and analyticity of free boundaries.

Findings

Proves optimal regularity of minimizers.

Establishes $C^{1,eta}$ regularity of free boundaries.

Shows analyticity of free boundaries for minimizers.

Abstract

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_{\Omega}\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0<p<1, $$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,\alpha}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity.