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Singular perturbations in fractional Sobolev spaces

Lecturer: 
Course Type: 
PhD Course
Academic Year: 
2024-2025
Period: 
October - February
Duration: 
40 h
Description: 
Singular perturbations are a classical way to tackle problems where either a solution is not ensured by a lack of coerciveness, or there are too many solutions, typically due to a lack of strict convexity.
In a variational setting, the perturbation is a higher-order strictly convex term multiplied by a small coefficient. The prototypical (well-known) example is a first-order perturbation of a non-convex term depending on a function u (typically a “double-well). Depending on the context this is known as a van der Waals, Cahn-Hiliard, or Modica-Mortola type functional.  The first-order term rules out discontinuous solutions, and as the parameter tends to 0 the gradient of the solutions tend to concentrate on an area-minimizing hypersurface. In the context of fractional Sobolev spaces, the same behaviour can be achieved by replacing the first-order term with a Gagliardo seminorm of order s with 1/2<s<1 (Alberti-Bellettini/Savin-Valdinoci theorem), while the case s=1/2 is more singular. Recent work on higher-order perturbations allows to replace the first-order term with a term of order k, with k integer larger than 2 (Fonseca and Mantegazza, Brusca, Donati and Solci). In those works a key ingredient is the use of interpolation inequalities, that allow one to estimate derivatives of intermediate order with the function and its k-th derivatives. 
 
With these results in mind the topics of the course will be as follows.
1) Brief recap of first-order perturbations of double-well problems;
2) A brief introduction to fractional variational problems;
3) Fractional-order perturbations of double-well problems;
4) The critical case s=1/2;
5) Higher-(integer)-order perturbations of double-well problems - integer perturbations;
6) Higher-order fractional Sobolev spaces;
7) Higher-(fractional)-order perturbations of double-well problems - fractional perturbations;
8) Degenerate problems at infinity. Free-discontinuity problems. The Blake-Zisserman and Perona-Malik functionals.
9) Singular perturbations of the Blake-Zisserman and Perona-Malik functionals - integer case;
10) Singular perturbations of the Blake-Zisserman and Perona-Malik functionals - fractional case

Schedule:

  • Tuesday, November 5 from 2:00 PM to 4:00 PM, room 128-129
  • Wednesday, November 6 from 11:00 AM to 1:00 PM, room 133
  • Friday, November 8 from 11:00 AM to 1:00 PM, room 136
  • Monday, November 11 from 11:00 AM to 1:00 PM, room 133
  • Monday, November 25 from 11:00 AM to 1:00 PM, room 133
  • Tuesday, November 26 from 2:00 PM to 4:00 PM room 133
  • Thursday, November 28 from 11:00 AM to 1:00 PM room 134
Next Lectures: 
Friday, November 8, 2024 - 11:00 to 13:00
Monday, November 11, 2024 - 11:00 to 13:00
Monday, November 25, 2024 - 11:00 to 13:00
Tuesday, November 26, 2024 - 14:00 to 16:00
Thursday, November 28, 2024 - 11:00 to 13:00

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