Energy Methods for Free Boundary Problems

1 Localized Solutions of Nonlinear Stationary Problems.- 1 Introduction.- 2 Second-order elliptic equations.- 3 The weighted diffusion/absorption balance.- 4 Anisotropic equations: Diffusion/absorption balance.- 5 Systems of second-order elliptic equations.- 6 Higher-order elliptic equations.- 7 Bibliographical notes and open problems.- 2 Stabilization in a Finite Time to a Stationary State.- 1 Introduction.- 2. Second-order parabolic equations.- 3 The weighted diffusion-absorption balance.- 4 The Cauchy problem.- 5 Equations with nonpower and isotropic nonlinearities.- 6 Systems of equations of combined type.- 7 Higher-order parabolic equations and other applications.- 8 Bibliographical notes and open problems.- 3 Space and Time Localization in Nonlinear Evolution Problems.- 1 Introduction.- 2 General second-order equations.- 3 The waiting time property.- 4 Shrinking of supports and formation of a dead core.- 5 Equations with nonhomogeneous absorption terms.- 6 Equations with anisotropic nonlinearities.- 7 Systems of parabolic equations.- 8 Higher-order parabolic equations.- 9 Bibliographical notes.- 4 Applications to Problems in Fluid Mechanics.- 1 Introduction.- 2 The balance laws of fluid mechanics.- 3 Stationary problems of gas dynamics with free boundaries.- 4 Two-phase filtration of immiscible incompressible fluids.- 5 Flows of gas with density-dependent viscosity.- 6 Viscous-elastic media.- 7 Flows of nonhomogeneous non-Newtonian fluid.- 8 Boundary layers in dilatant fluid.- 9 Boussinesq system involving nonlinear thermal diffusion.- 10 Simultaneous motion in the surface channel and the underground water.- 11 Solute transport through a porous medium with micro and macro structures.- 12 A quasilinear degenerate system arising in semiconductor theory.- 13 Blowup in solutions of the thermistor problem.- The function spaces.- Elementary inequalities.- 2.1 Algebraic inequalities.- 2.2 Integral inequalities.- 3 Embedding theorems.- 3.1 Interpolation inequalities.- 3.2 Anisotropic function spaces.- References.