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Regularity of a vector valued two phase free boundary problems

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  • Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq2$ and $\Sigma$ be a $q$ dimensional smooth submanifold of $\mathbb{R}^{m}$ with $0 \leq q < m$. We use $\mathcal{M}_{\Omega,\Sigma}$ to denote the collection of all pairs of $(A,u) $ such that $A\subset\Omega$ is a set of finite perimeter and $u\in H^{1}\left( \Omega,\mathbb{R}^{m}\right) $ satisfies \[ u\left( x\right) \in\Sigma\text{ a.e. }x\in A. \] We consider the energy functional \[ E_{\Omega}\left( A,u\right) =\int_{\Omega}\left\vert \nabla u\right\vert ^{2}+P_{\Omega}\left( A\right) , \] defined on $\mathcal{M}_{\Omega,\Sigma}$, where $P_{\Omega}\left( A\right) $ denotes the perimeter of $A$ inside $\Omega$. Let $\left( A,u\right) $ be a local energy minimizer. Our main result is that when $n\leq7$, $u$ is locally Lipschitz and the free boundary $\partial A$ is smooth in $\Omega$.
    Mathematics Subject Classification: Primary: 35R35; Secondary: 35J20, 76A15.


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