Colloquium on the occasion of Professor Frehse’s 75th birthday

Date: April 5, 2019

Venue: Lipschitz lecture hall, Mathematics Center, Endenicher Allee 60, 53115 Bonn

In this talk we consider a large class of non-uniformly elliptic variational problems and discuss optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the data. The analysis covers the main model cases of variational integrals of anisotropic growth, but also of fast growth of exponential type investigated in recent years. The regularity criteria are established by potential theoretic arguments, involve natural limiting function spaces on the data, and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. The results presented in this talk are part of a joined project with Giuseppe Mingione (Parma).

We give an overview of recent regularity results for nonlocal equations. These equations are driven by integro-differential operators with differentiability order between 0 and 2. We explain analogies and differences with respect to the theory of partial differential equations of second order.

With "signed pressure" we mean a pressure which satisfies some pointwise sign condition. We refer to a paper of Seregin-Sverak about weak solutions (v,q) to Navier-Stokes-equations with a nonnegative pressure q. Roughly, it states that this condition implies full regularity for the velocity field v, and q. We present alternative  sign conditions for the pressure, which allow a short proof for the regularity of (v,q). This can be used also as a criterium for improved integrability exponents for p-fluids.

*In German, but with english slides.