INTRODUCTION TO SOME ASPECTS OF FREE SURFACE PROBLEMS

Abstract

This chapter presents an introduction to some aspects of free surface problems. It reviews the connections between a few free boundary problems and a few classes of variational inequalities (V.I.), depicting how these connections can be useful. The chapter presents the penalty method, which gives an approximation procedure for the solution of the V.I. and gives an error estimate. The chapter highlights the techniques of V.I., together with asymptotic methods of Sanchez−Palencia, de Giorgi−Spagnolo, Babuškato to obtain an asymptotic result for the solution of a free boundary problem in a very inhomogeneous material. It focuses on the advantage of the formulation as a V.I., in a strong form or in a weak form. The chapter discusses penalty on the Stefan's problem, the free boundary problem in a composite material, and the homogenized free boundary problem.