# On the Lipschitz regularity of solutions of a minimum problem with free boundary

### Aram L. Karakhanyan

Australian National University, Canberra, Australia

## Abstract

In this article under assumption of "small" density for negativity set, we prove local Lipschitz regularity for the one phase minimization problem with free boundary for the functional

$\mathcal E_p(v,\Omega)=\int_\Omega|\nabla v|^p+\lambda^p_1\X{u\leq0}+\lambda^p_2\X{u>0},\hspace{3mm} 1<p<\infty,$

where $\lambda_1, \lambda_2$ are positive constants so that $\Lambda=\lambda_1^p-\lambda_2^p<0$, $\chi_{D}$ is the characteristic function of set $D$, $\Omega\subset\Rn$ is (smooth) domain and minimum is taken over a suitable subspace of $W^{1,p}(\Omega)$.

## Cite this article

Aram L. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary. Interfaces Free Bound. 10 (2008), no. 1, pp. 79–86

DOI 10.4171/IFB/180