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2013, 2013(special): 365-374. doi: 10.3934/proc.2013.2013.365

## Regularity of a vector valued two phase free boundary problems

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15024, United States

Received  September 2012 Revised  December 2012 Published  November 2013

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 $\Omega$.
Citation: Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365
##### References:
 [1] I. Athanasopoulos, L. A. Caffarelli, C. Kenig, and S. Salsa., An area-Dirichlet integral minimization problem., {\em Comm. Pure Appl. Math.}, (2001), 479.   Google Scholar [2] Lawrence C. Evans and Ronald F. Gariepy., Measure theory and fine properties of functions., Studies in Advanced Mathematics. CRC Press, (1992).   Google Scholar [3] P. G. De Gennes., The physics of liquid crystals., Studies in Advanced Mathematics. Clarendon Press, (1974).   Google Scholar [4] Huiqiang Jiang., Analytic regularity of a free boundary problem., {\em Calc. Var. Partial Differential Equations}, (2007), 1.   Google Scholar [5] Huiqiang Jiang and Christopher Larsen., Analyticity for a two dimensional free boundary problem with volume constraint., Preprint., ().   Google Scholar [6] Huiqiang Jiang, Christopher J. Larsen, and Luis Silvestre., Full regularity of a free boundary problem with two phases., {\em Calc. Var. Partial Differential Equations}, (2011), 3.   Google Scholar [7] Huiqiang Jiang and Fanghua Lin., A new type of free boundary problem with volume constraint., {\em Comm. Partial Differential Equations}, (2004), 5.   Google Scholar [8] Paolo Tilli., On a constrained variational problem with an arbitrary number of free boundaries., {\em Interfaces Free Bound.}, (2000), 201.   Google Scholar

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