A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,α.

Luis A. Caffarelli

Revista Matemática Iberoamericana (1987)

  • Volume: 3, Issue: 2, page 139-162
  • ISSN: 0213-2230

Abstract

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This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We plan to do so by:(a) constructing rather weak generalized solutions of the free-boundary problems,(b) showing that the free boundary of such solutions have nice measure theoretical properties (i.e., finite (n-1)-dimensional Hausdorff measure and the associated differentiability properties),(c) showing that near a flat point free-boundaries are Lipschitz graphs, and(d) showing that Lipschitz free boundaries are really C1,α.

How to cite

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Caffarelli, Luis A.. "A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,α.." Revista Matemática Iberoamericana 3.2 (1987): 139-162. <http://eudml.org/doc/39343>.

@article{Caffarelli1987,
abstract = {This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We plan to do so by:(a) constructing rather weak generalized solutions of the free-boundary problems,(b) showing that the free boundary of such solutions have nice measure theoretical properties (i.e., finite (n-1)-dimensional Hausdorff measure and the associated differentiability properties),(c) showing that near a flat point free-boundaries are Lipschitz graphs, and(d) showing that Lipschitz free boundaries are really C1,α.},
author = {Caffarelli, Luis A.},
journal = {Revista Matemática Iberoamericana},
keywords = {Problemas de frontera libre; Función armónica; Soluciones; existence; two-phase free-boundary problem; Lipschitz free boundaries; regularity},
language = {eng},
number = {2},
pages = {139-162},
title = {A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,α.},
url = {http://eudml.org/doc/39343},
volume = {3},
year = {1987},
}

TY - JOUR
AU - Caffarelli, Luis A.
TI - A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,α.
JO - Revista Matemática Iberoamericana
PY - 1987
VL - 3
IS - 2
SP - 139
EP - 162
AB - This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We plan to do so by:(a) constructing rather weak generalized solutions of the free-boundary problems,(b) showing that the free boundary of such solutions have nice measure theoretical properties (i.e., finite (n-1)-dimensional Hausdorff measure and the associated differentiability properties),(c) showing that near a flat point free-boundaries are Lipschitz graphs, and(d) showing that Lipschitz free boundaries are really C1,α.
LA - eng
KW - Problemas de frontera libre; Función armónica; Soluciones; existence; two-phase free-boundary problem; Lipschitz free boundaries; regularity
UR - http://eudml.org/doc/39343
ER -

Citations in EuDML Documents

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  1. Fausto Ferrari, Teoremi di confronto di tipo Harnack per funzioni armoniche in domini con frontiera hölderiana
  2. L. Fornari, Regularity of the free boundary for non degenerate phase transition problems of parabolic type
  3. Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part III : existence theory, compactness, and dependence on X
  4. Claudia Lederman, Noemi Wolanski, Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem
  5. Eduardo V. Teixeira, A variational treatment for general elliptic equations of the flame propagation type : regularity of the free boundary
  6. Avner Friedman, Yong Liu, A free boundary problem arising in magnetohydrodynamic system
  7. Sandro Salsa, Viscosity Solutions of Two-Phase Free Boundary Problems for Elliptic and Parabolic Operators
  8. John L. Lewis, Kaj Nyström, Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

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