On the Lipschitz regularity of solutions of a minimum problem with free boundary (Q933106)

Summary: 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 \[ {\mathcal E}_p(v,\Omega)=\int_\Omega [|\nabla v|^p+ \lambda^p_1\chi_{\{u\leq0\}}+\lambda^p_2 \chi_{\{u>0\}}],\quad 1<p<\infty, \] 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 \(D\), \(\Omega\subset\mathbb R^n\) is (smooth) domain and minimum is taken over a suitable subspace of \(W^{1,p}(\Omega)\).