Regularity of solutions for a free boundary problem in two dimensions

TL;DR

This paper proves that in two-dimensional free boundary problems with measurable coefficients, minimizers are H"older continuous and exhibit linear growth near the free boundary, advancing understanding of solution regularity.

Contribution

It establishes regularity and growth properties of minimizers in two dimensions for a class of free boundary problems with minimal coefficient regularity.

Findings

Minimizers are H"older continuous in 2D.

Minimizers satisfy linear growth near the free boundary.

Results hold for coefficients only assumed bounded and measurable.

Abstract

We study the regularity of minimizers to the functional \[ J(w)=\int_{\Omega} a^{ij}w_iw_j + Q\chi_{\{w>0\}}, \] over a bounded domain $\Omega$ and among the class of nonnegative functions in $W^{1,2}(\Omega)$ with prescribed boundary data. We assume that the coefficients $a^{ij}$ are only bounded and measurable and satisfy an ellipticity in condition. In two dimensions we prove that minimizers are H\"older continuous on subdomains. We also prove that in two dimensions a minimizer $u$ satisfies a linear growth condition from above and below near the free boundary $\partial \{u>0\}$.