Differential and Integral Equations

Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators

Toyohiko Aiki

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Abstract

In this paper we discuss weak solutions of the multi-dimensional two-phase Stefan problem with the boundary condition including the subdifferential operator of the convex function on $\mathbb R$ so that the boundary condition is nonlinear and contains a multi-valued operator, in general. Kenmochi and Pawlow already established the uniqueness and existence of a solution to our problem in the sense of the vanishing viscosity solution. The purpose of this paper is to prove the uniqueness theorem for a solution defined in the usual variational sense. Our proof is due to the standard method in which the dual problem of the original problem plays a very important role.

Article information

Source
Differential Integral Equations, Volume 15, Number 8 (2002), 973-1008.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060781

Mathematical Reviews number (MathSciNet)
MR1895575

Zentralblatt MATH identifier
1017.35050

Subjects
Primary: 35R35: Free boundary problems
Secondary: 35K55: Nonlinear parabolic equations 80A22: Stefan problems, phase changes, etc. [See also 74Nxx]

Citation

Aiki, Toyohiko. Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators. Differential Integral Equations 15 (2002), no. 8, 973--1008. https://projecteuclid.org/euclid.die/1356060781


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