FIS3 StG: GEMS

Project GEMS focuses on regularity and rigidity questions arising in the field of partial differential equations and systems. Let us provide two examples to outline its scope.

• Critical and stationary points of polyconvex integrands:

Polyconvex integrands are among the most important and versatile classes of elliptic energies in the vectorial Calculus of Variations. Among them, the prototypical example is the area integrand on graphs of codimension higher than one. It is known that minimizers of these energies are (partially) regular, while the same property spectacularly fails for critical points. The same question is still open for stationary points, i.e., maps which are critical with respect to both outer and domain variations. As the relation with the area integrand suggests, this question is highly relevant in connection to regularity issues in Geometric Measure Theory.

• Equations displaying degenerate ellipticity:

in codimension one, i.e., for scalar equations arising from uniformly convex functionals, the celebrated De Giorgi-Nash-Moser theorem shows regularity of minima. This has been extended in several ways and, by now, even regularity results for more general operators such as the p-Laplacian are classical. It is, however, of interest to understand how much further such regularity results can be pushed, both in the direction of concentration issues, for example, by considering very weak solutions to the p-Laplace equation, or of oscillation issues, by considering Lipschitz solutions to equations with strict but unquantified ellipticity.

The main viewpoint adopted by this project is that of differential inclusions: most partial differential equations can be recast efficiently in terms of this language, and this unlocks a vast array of tools to study them. In turn, an important part of this project is also devoted to sharpen and expand such tools, for instance by considering inclusions associated to general linear operators with constant coefficients, which are related to fundamental PDEs and systems in higher dimensions.

Generally speaking, the main underlying question is whether the given differential inclusion is rigid or flexible:

•  Rigidity, or ellipticity, amounts to showing that every solution to the given differential inclusion enjoys higher regularity. To do so, one typically shows elliptic estimates through compensated compactness, tools from the theory of quasiregular maps, topological methods, etc.;

•  Flexibility, on the other hand, allows the use of convex integration, which in this context is the art of finding micro-structures inside the inclusion set. Convex integration methods have been successfully employed in both directions outlined above.

For an essential bibliography to get familiar with the topics of this project, press the button below.

• Riccardo Tione (PI)

One Phd position and several postdoc positions will be available in the course of the next five years. If you wish to express interest in working on this project, please send an email to riccardo.tione@unito.it, attaching your CV.