# Regularity for Non-Linear PDEs

## Partially Supported by ERC grant

31 August 2009 - 27 September 2009

# Aims

The period aims at gathering a large number of leading experts in the field of regularity theory for non-linear partial differential equations and systems of elliptic and parabolic type, and for integral functionals of the Calculus of Variations.

In the last years there has been a renewed interest in regularity problems for solutions to pde, and a large number of studies has been devoted to such Amongst the others the following topics have undergone considerable developments:

-- Regularity of solutions to geometric problems: conformally invariant geometric problems governed by hidden conservation laws and compensation phenomena; related regularity theorems for general elliptic systems with critical growth (see for instance the recent work of Riviere on harmonic maps and the Willmore functional).

-- New regularity approaches to harmonic and bi-harmonic mappings.

-- New estimates for the singular sets of solutions to vectorial problems: solutions to elliptic and parabolic systems, and minimizers of integral functionals of the Calculus of Variations develop singularities; the study of the structure properties of the singular set is a related issue of great current interest.

-- Evolution problems related to the curvatures: mean and inverse-mean curvature evolution, Ricci flow and related entropy estimates.

-- Non-linear potential theory and the geometry and regularity problems for solutions to degenerate equations of $p$-Laplacen type; related non-uniformly elliptic and parabolic operators, and problems with non-standard growth conditions.

-- Regularity of solutions to equations on manifolds and Lie groups: the prototype in this case is given by the $p$-Laplacean operator in the Heisenberg group, whose regularity properties are little understood.

-- Regularity for highly degenerate equations as the $\infty$-Laplace equation.

-- Regularity for fully non-linear elliptic equations and connections to free boundary problems and optimal transportation problems.

-- Regularity for elliptic and parabolic problems with fractional order of differentiation and viscosity solutions to fully non-linear

-- Non-linear parabolic equations with various forms of degeneracy.

-- Regularity for problems coming from Fluid-Dynamics, Navier-Stokes equations.

The proposed research period, lasting four weeks, aims at touching all the foregoing points, and proposing an up-dated review of the state of the art for them, offering a number of mini-courses (eight minicourses of eight hours each, for approximative three hours of lectures per day), plus a number of seminars (approximatively 23).

The presence of leading experts for a substantial number of days, interacting also with the audience, will be one of goals of the organizers.