In the first part of the course I will review the main metric, embedding and structure theorem about Sobolev spaces (depending on the audience more or less in depth), and study the corresponding weak convergence. I will then study lowersemicontinuity conditions for local functionals (that is, integral functionals depending on the weak gradient), described by convexity conditions (convexity, polyconvexity, quasiconvexity, rank1convexity), and apply them to obtain solutions of minimum problems by the Direct Method of the Calculus of Variations. I will discuss relaxation results for problems involving nonlowersemicontinuous functionals and state some open problems.In the second part of the course, I will introduce (Riesz) fractional gradients and the related fractional Sobolev spaces. I will prove structure properties of such spaces and highlight differences and analogies with the usual weak gradient. I will define a weak convergence and study weak lowersemicontinuous properties for integral functionals depending on fractional gradients.In the third part of the course I will consider a finitedifference type approach to Sobolev spaces. To that end I will consider double integrals concentrating on the diagonal and prove asymptotic structure properties and Poincaré inequalities. As a result, I will show that local minimum problems on Sobolev spaces can be approximated by problems on Lebesgue spaces involving double integrals. I will discuss lowersemicontinuity and relaxation problems for double integrals and state some open problems.
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Local and nonlocal variational problems in Sobolev spaces
Lecturer:
Course Type:
PhD Course
Academic Year:
20222023
Period:
October  March
Duration:
50 h
Description:
Research Group:
Location:
A134
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