## Optimization problems in calculus of variations

More and more problems arising from Applied Mathematics, Economics and Engineering, but also from real life, have been studied by models in calculus of variations.The optimization problems concerning the spectrum of the Laplacian are part of the classical problems in the analysis and are currently receiving a lot of attention. For example, a classical problem consistsin rigidifyinga membrane, under the action of an exterior load f and fixed at its boundary. A generalization consistsin adding a one-dimensional reinforcement in the most efficient way. The reinforcement will be described by a one-dimensional set which varies in a suitable class of admissible choices. Another one consists in studying spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue. Some classical estimates involving the torsional rigidity and the principal frequency of a convex domain are availablesince the ‘60 but it is interesting to extend it to a class of functionals related to some anisotropic non linear operators. Results concerning the approximation of the BV-norm by non-local functionals have been studied recently by several authors. Also, the characterization of Sobolev spaces (included higher order) and of the space BV, the space of functions of higher order bounded variation in terms of BMO-type seminorms, receiveda lot of attention. Spaces of this kind have applications in mathematical imaging in the setting of isotropic and anisotropic variants of the TV-model. Total variation minimizing models (introduced by Rudin, Osher and Fatemi in 1992) have been employed in a wide variety of image restoration problems, quickly becoming one of the most active research topics in image processing and computer vision. The most basic and, nonetheless, fundamental, image restoration problem is denoising, that is, preserving the most significant features of an image, such as those most easily detected by the human visual system, while removing the noise.The free boundary problems are a special type of boundary value problems, in which the domain, where the PDE is solved, depends on the solution of the boundary value problem. A free boundary problem of particularrelevance for the theory is the so-called one-phase-Bernoulli problem, which was the object of numerous studies in the last 40 years; it also motivated the introduction of several new tools and the development of new regularity techniques. Free boundary problems with two and more phases are often used to describe models in different areas of Physics, Engineering and Life Sciences, for instance in Fluid Dynamics (Bernoulli free boundary problems), Dynamics of Populations (optimal partitions problems), Mechanics and Phase Transition (obstacle problems). The different phases are called segregated if they occupy different space regions; segregation occurs for instance in the two-phase Bernoulli problem, the two-phase obstacle problem and optimal partitions problems.

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[Ultimo aggiornamento: 04/04/2023 11:17:43]