Mathematics

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