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＜E-Book＞
Dual Variational Approach to Nonlinear Diffusion Equations / by Gabriela Marinoschi
(PNLDE Subseries in Control. ISSN:27317374 ; 102)

Edition	1st ed. 2023.
Publisher	(Cham : Springer Nature Switzerland : Imprint: Birkhäuser)
Year	2023
Language	English
Size	XVIII, 212 p : online resource
Authors	*Marinoschi, Gabriela author SpringerLink (Online service)
Subjects	LCSH:Differential equations LCSH:System theory LCSH:Control theory LCSH:Operator theory LCSH:Mathematical optimization LCSH:Calculus of variations FREE:Differential Equations FREE:Systems Theory, Control FREE:Operator Theory FREE:Calculus of Variations and Optimization
Notes	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 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 models tovarious real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well HTTP:URL=https://doi.org/10.1007/978-3-031-24583-1

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