Variational formulas on Lipschitz domains

1995/01/01 by Alan R. Elcrat, Kenneth G. Miller · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research

paper · pdf · doi:10.1090/s0002-9947-1995-1285987-2

Abstract

A rigorous treatment is given of variational formulas for solutions of certain Dirichlet problems for the Laplace operator on Lipschitz domains under interior variations. In particular we extend well-known variational formulas for the torsional rigidity and for capacity from the class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">C¹</mml:annotation> </mml:semantics> </mml:math> </inline-formula> domains to the class of Lipschitz domains. A motivation for this work comes from the use of variational methods in the study of Prandtl-Batchelor flows in fluid mechanics.

Citations

• Variational problems with two phases and their free boundaries

Cited by

• On semilinear equations with free boundary conditions on stratified Lie groups
• Analysis of a free boundary problem on stratified Lie group
• Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

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