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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:
  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  Lawrence C. Evans and Ronald F. Gariepy., Measure theory and fine properties of functions., Studies in Advanced Mathematics. CRC Press, (1992). Google Scholar  P. G. De Gennes., The physics of liquid crystals., Studies in Advanced Mathematics. Clarendon Press, (1974).   Google Scholar  Huiqiang Jiang., Analytic regularity of a free boundary problem., {\em Calc. Var. Partial Differential Equations}, (2007), 1. Google Scholar  Huiqiang Jiang and Christopher Larsen., Analyticity for a two dimensional free boundary problem with volume constraint., Preprint., ().   Google Scholar  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  Huiqiang Jiang and Fanghua Lin., A new type of free boundary problem with volume constraint., {\em Comm. Partial Differential Equations}, (2004), 5. Google Scholar  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:
  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  Lawrence C. Evans and Ronald F. Gariepy., Measure theory and fine properties of functions., Studies in Advanced Mathematics. CRC Press, (1992). Google Scholar  P. G. De Gennes., The physics of liquid crystals., Studies in Advanced Mathematics. Clarendon Press, (1974).   Google Scholar  Huiqiang Jiang., Analytic regularity of a free boundary problem., {\em Calc. Var. Partial Differential Equations}, (2007), 1. Google Scholar  Huiqiang Jiang and Christopher Larsen., Analyticity for a two dimensional free boundary problem with volume constraint., Preprint., ().   Google Scholar  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  Huiqiang Jiang and Fanghua Lin., A new type of free boundary problem with volume constraint., {\em Comm. Partial Differential Equations}, (2004), 5. Google Scholar  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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