Dual variational approach to nonlinear diffusion equations (Q6044566)

From the author's preface: ``The aim of this book is to present a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. This formulation issued from physics and initially established as a principle by Brezis and Ekeland in 1976 was continued in some other authors' works after the 1980s and was systematically treated and developed in a unifying framework by Ghoussoub in a monograph in 2009. Our purpose is to show how this method turns out to be a convenient tool for proving the existence of the solutions to nonlinear diffusion equations when other methods may fail. This is the case of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. In all these situations which are difficultly treated or may be not treated by other methods, this formulation, essentially employing the dual Legendre-Fenchel relations, proves to be an elegant and versatile technique involving knowledge from the theory of maximal monotone operators, semigroup theory, convex functions, and their subdifferentials. Besides its effectiveness in proving the existence of solutions to nonlinear partial differential equations, this technique is useful in the treatment of inverse problems and optimal control problems which are approached in this book as well.'' Let us mention in addition that the book contains 7 chapters, as follows: 1. Nonlinear diffusion equations with slow and fast diffusion; 2. Weakly coercive nonlinear diffusion equations; 3. Nonlinear diffusion equations with a noncoercive potential; 4. Nonlinear parabolic equations in divergence form with Wentzell boundary conditions; 5. A nonlinear control problem in image denoising; 6. An optimal control problem for a phase transition model; 7. Appendix (including some well-known results of functional analysis, operator theory, and convex analysis, without proofs), plus References and an Index. The book might be useful for graduate students and researchers working on related topics.