Dual Variational Approach to Nonlinear Diffusion Equations

This monograph presents a dual variational formulation of solutions to nonlinear diffusion equations,which is a useful tool for proving the existence of solutions when other methods fail. It simplifies the proof of the existence of minimizers and determines the first-order conditions of optimality. The formulation is illustrated with specific diffusion equations representing real-world physical processes,and it is also applicable to inverse problems and optimal control problems.

Format: Hardback
Length: 212 pages
Publication date: 29 March 2023
Publisher: Birkhauser Verlag AG

This monograph delves into a novel approach to solving nonlinear diffusion equations with general nonlinearities by utilizing a dual variational formulation. The author showcases how this method can be employed as a powerful tool for proving the existence of solutions, particularly in cases where other methods may fail, such as when dealing with evolution equations with nonautonomous operators, low regular data, or singular diffusion coefficients. By transforming the original problem into a minimization problem, it is simplified into an optimal control problem with a linear state equation. This streamlined approach makes it easier to establish the existence of minimizers and determine the first-order conditions of optimality.

The dual variational formulation is exemplified through specific diffusion equations with general nonlinearities derived from potentials possessing varying strengths or weaknesses. These equations can serve as mathematical models for a wide range of real-world physical processes. Additionally, the text explores inverse problems and optimal control problems, recognizing the usefulness of this technique in their treatment as well.

Overall, this monograph offers a valuable contribution to the field of mathematical physics by providing a novel and effective method for solving nonlinear diffusion equations with general nonlinearities. Its application to a diverse range of physical scenarios demonstrates its versatility and potential for advancing our understanding of complex systems.

Weight: 518g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031245824
Edition number: 1st ed. 2023