January & February  2004, 10(1&2): 397-422. doi: 10.3934/dcds.2004.10.397

On the free boundary regularity theorem of Alt and Caffarelli

1. 

Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States

2. 

Department of Mathematics, University of Washington, Seattle, Washington 98195–4350, United States

Received  May 2002 Revised  April 2003 Published  October 2003

In this note we discuss a slight generalization of the following result by Alt and Caffarelli: if the logarithm of the Poisson kernel of a Reifenberg flat chord arc domain is Hölder continuous, then the domain can be locally represented as the area above the graph of a function whose gradient is Hölder continuous. In this note we show that if the Poisson kernel of an unbounded Reifenberg flat chord arc domain is 1 a.e. on the boundary then the domain is (modulo rotation and translation) the upper half plane. This result plays a key role in the study of regularity of the free boundary below the continuous threshold.
Citation: 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
[1]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[2]

Steven G. Krantz and Marco M. Peloso. New results on the Bergman kernel of the worm domain in complex space. Electronic Research Announcements, 2007, 14: 35-41. doi: 10.3934/era.2007.14.35

[3]

Dian Palagachev, Lubomira G. Softova. Quasilinear divergence form parabolic equations in Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1397-1410. doi: 10.3934/dcds.2011.31.1397

[4]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015

[5]

Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39

[6]

Alice B. Tumpach, Stephen C. Preston. Quotient elastic metrics on the manifold of arc-length parameterized plane curves. Journal of Geometric Mechanics, 2017, 9 (2) : 227-256. doi: 10.3934/jgm.2017010

[7]

Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4791-4804. doi: 10.3934/dcds.2015.35.4791

[8]

Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617

[9]

Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055

[10]

Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199

[11]

François Bolley, Arnaud Guillin, Xinyu Wang. Non ultracontractive heat kernel bounds by Lyapunov conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 857-870. doi: 10.3934/dcds.2015.35.857

[12]

Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations & Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399

[13]

Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763

[14]

Boris Kolev. Poisson brackets in Hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 555-574. doi: 10.3934/dcds.2007.19.555

[15]

Said Hadd, Rosanna Manzo, Abdelaziz Rhandi. Unbounded perturbations of the generator domain. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 703-723. doi: 10.3934/dcds.2015.35.703

[16]

Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4375-4400. doi: 10.3934/dcds.2013.33.4375

[17]

Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783

[18]

Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333

[19]

Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915

[20]

Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]