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 lower-semicontinuity conditions for local functionals (that is, integral functionals depending on the weak gradient), described by convexity conditions (convexity, polyconvexity, quasiconvexity, rank-1-convexity), 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 non-lower-semicontinuous 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 lower-semicontinuous properties for integral functionals depending on fractional gradients.In the third part of the course I will consider a finite-difference 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 lower-semicontinuity and relaxation problems for double integrals and state some open problems.
You are here
Local and nonlocal variational problems in Sobolev spaces
Lecturer:
Course Type:
PhD Course
Academic Year:
2022-2023
Period:
October - March
Duration:
50 h
Description:
Research Group:
Location:
A-134
Openings
- Call of interest for positions in mathematics at SISSA
- Public Calls for Professors
- Temporary Professors/Researchers/Visiting Professors
- SISSA Mathematical Fellowships
- Post-Lauream Fellowships
- Research Training Fellowships
- Marie Sklodowska-Curie Grants
- Open positions in MathLab
- Post Doctoral Fellowships
- PhD Scolarships
- SIS Fellowships
- Undergraduate Fellowships
- Postgraduate Fellowships
- MSc in Mathematics
- MSc in Data Science and Scientific Computing (DSSC)
- Professional Master Courses
- SISSA Mathematics Medals
Upcoming events
-
Jonathan Husson
Random matrices, large deviations and spherical integral
Monday, December 11, 2023 - 16:00 to 17:00
-
Gabriele Benedetti
From symplectic capacities to periodic Hamiltonian flows
Thursday, December 14, 2023 - 14:00
Today's Lectures
-
Stefano Piani
09:00 to 11:00
-
09:00 to 11:00
-
11:00 to 13:00
-
Daniele Semola
14:00 to 16:00
-
14:00 to 16:00
-
16:00 to 18:00
Recent publications
-
S. Carano,Relaxed area of $0$-homogeneou...
-
S. Salavatidezfouli; S. Hajisharifi; M. Girfoglio; G. Stabile; G. Rozza,Applicable Methodologies for t...
-
G. Bellettini; S. Carano; R. Scala,Relaxed area of graphs of piec...
-
S. Bianchini; M. Zizza,Properties of Mixing BV Vector...