Regularity of a vector valued two phase free boundary problems

Huiqiang Jiang

doi:10.3934/proc.2013.2013.365

Article Contents

2013, Volume 2013: 365-374. Doi: 10.3934/proc.2013.2013.365 open access

This issue Previous Article Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization Next Article Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid

Regularity of a vector valued two phase free boundary problems

Huiqiang Jiang^1,

1.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15024

Received: September 2012

Revised: December 2012

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

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 $\Omega$.

Keywords:

Mathematics Subject Classification: Primary: 35R35; Secondary: 35J20, 76A15.

Citation:

Related Papers

Cited by

References

[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.