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

show all references

##### 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
  Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591  Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431  Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025  Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751  Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397  Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013  Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103  Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429  Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006  Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025  Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386  Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481  Samuel Amstutz, Antonio André Novotny, Nicolas Van Goethem. Minimal partitions and image classification using a gradient-free perimeter approximation. Inverse Problems & Imaging, 2014, 8 (2) : 361-387. doi: 10.3934/ipi.2014.8.361  Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447  Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017  Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673  Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799  Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179  Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239  Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233

Impact Factor: