A generalized Stefan problem in several space variables

Abstract

Abstract.A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.

Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation

“…The authors show the existence of a solution and provide numerical experiments. Niezgódka and Pawlow [21] discuss results on weak solutions of multidimensional Stefan problems in enthalpy formulation. In the companion paper [20], they use these results to show existence of optimal controls.…”

A note on the approximation of free boundaries by finite element methods

“…Hère iï is the Heaviside graph and c is a strictly increasing Lipschitz continuous function. It is well-known that there is a unique solution u (physically, the température) ; moreover it has the global regularity properties [11,17,18,19,23,31] : assume u 0 := J~1(y o ) e C oa (Q) and ƒ e C 04 (R), then (4 * 2) we C°(ë) n L°°(0, T\H\Q)) n H^O, T; L 2 (Q)).…”

Weak solutions for Falk's model of shape memory alloys

“…By using the dual problem instead of the original problem we prove uniqueness. This method can be found in Chapter 3 of the monograph [7], and was improved and applied to multi-dimensional Stefan problems with non-linear boundary conditions by NiezgoH dka and Pawlow [8]. Also, the author established the uniqueness for a multi-dimensional Stefan problem with the boundary condition described by maximal monotone operators in [1].…”