Article Contents
Article Contents

Regularity of a vector valued two phase free boundary problems

• 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.

 Citation:

•  [1] I. Athanasopoulos, L. A. Caffarelli, C. Kenig, and S. Salsa., An area-Dirichlet integral minimization problem. Comm. Pure Appl. Math., 54(4):479-499, 2001. [2] Lawrence C. Evans and Ronald F. Gariepy., Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [3] P. G. De Gennes., The physics of liquid crystals. Studies in Advanced Mathematics. Clarendon Press, Oxford, 1974. [4] Huiqiang Jiang., Analytic regularity of a free boundary problem. Calc. Var. Partial Differential Equations, 28(1):1-14, 2007. [5] Huiqiang Jiang and Christopher Larsen., Analyticity for a two dimensional free boundary problem with volume constraint. Preprint. [6] Huiqiang Jiang, Christopher J. Larsen, and Luis Silvestre., Full regularity of a free boundary problem with two phases. Calc. Var. Partial Differential Equations, 42(3-4):301-321, 2011. [7] Huiqiang Jiang and Fanghua Lin., A new type of free boundary problem with volume constraint. Comm. Partial Differential Equations, 29(5-6):821-865, 2004. [8] Paolo Tilli., On a constrained variational problem with an arbitrary number of free boundaries. Interfaces Free Bound., 2(2):201-212, 2000.
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