This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical modelsto various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.

Gabriela Marinoschi is a senior scientific researcher with Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy and full member of the Romanian Academy. Her research interests focus on the analysis and control of evolution equations in infinite dimensional spaces and include the application of variational and semigroup methods as well as the control techniques to mathematical models based on partial differential equations, especially for those describing physical and biological processes.

Introduction.- Nonlinear Diffusion Equations with Slow and Fast Diffusion.- Weakly Coercive Nonlinear Diffusion Equations.- Nonlinear Diffusion Equations with a Noncoercive Potential.- Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary Conditions.- A Nonlinear Control Problem in Image Denoising.- An Optimal Control Problem for a Phase Transition Model.- Appendix.- Bibliography.- Index.