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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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  • [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.


    Lawrence C. Evans and Ronald F. Gariepy., Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.


    P. G. De Gennes., The physics of liquid crystals. Studies in Advanced Mathematics. Clarendon Press, Oxford, 1974.


    Huiqiang Jiang., Analytic regularity of a free boundary problem. Calc. Var. Partial Differential Equations, 28(1):1-14, 2007.


    Huiqiang Jiang and Christopher Larsen., Analyticity for a two dimensional free boundary problem with volume constraint. Preprint.


    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.


    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.


    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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