Some New Monotonicity Theorems with Applications to Free Boundary Problems

2002/03/01 by Luis A. Caffarelli, Luis Caffarelli, David Jerison +1 · 17 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Contact Mechanics and Variational Inequalities

paper · doi:10.2307/3062121

Abstract

The monotonicity theorem of Alt, Caffarelli, and Friedman [ACF, Lemma 5.1] plays a central role in the existence and regularity theory of two-phase free boundary problems. It has been extended, for example, to variable coefficient operators [C3, Lemma 1] and to eigenvalue problems [FL]. In this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right-hand side of the equation need not vanish at the free boundary. Of particular interest is the case in which Au takes two different constant values on the set where u > 0 and on the set where u < 0. The Prandtl-Batchelor

Cited by

• On semilinear equations with free boundary conditions on stratified Lie groups
• Analysis of a free boundary problem on stratified Lie group
• Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case
• Uniform Hölder Estimates in a Class of Elliptic Systems and Applications to Singular Limits in Models for Diffusion Flames
• Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients
• Extreme internal waves: gravity currents and overturning fronts
• Regularity for degenerate two-phase free boundary problems
• A monotonicity formula with application to the relativistic Brillouin electron flow
• Higher critical points in a free boundary problem
• A free boundary problem for the localization of eigenfunctions
• Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds
• A local parabolic monotonicity formula on Riemannian manifolds
• Strong competition versus fractional diffusion: the case of\n Lotka-Volterra interaction
• On a fractional Alt-Caffarelli-Friedman-type monotonicity formula
• Borderline regularity in singular free boundary problems
• A variational problem for the spatial segregation of reaction--diffusion systems
• Two- and Multi-phase Quadrature Surfaces

Related

• Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds
• Regularity of higher order in two-phase free boundary problems
• A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula
• A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two
• A monotonicity formula for a classical free boundary problem