L6

The Maxwell-Klein-Gordon equation models the interaction of an electromagnetic field with a charged particle field. We discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" Bahouri-Gérard type profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of critical wave maps. This is joint work with Joachim Krieger.

Level sets of solutions to elliptic and parabolic problems are often much more regular than the equation suggests. I will discuss partial analyticity and consequences for level sets, the regularity of solutions to elliptic PDEs in some limit cases, and the regularity of flow lines for bounded stationary solutions to the Euler equation. This is joint work with Nikolai Nadirashvili.

We address regularity properties of (vector-valued) weak solutions to quasilinear elliptic systems, for the special situation that the inhomogeneity grows naturally in the gradient variable of the unknown (which is a setting appearing for various applications). It is well-known that such systems may admit discontinuous and even unbounded solutions, when no additional structural assumption on the inhomogeneity or on the leading elliptic operator or on the solution is imposed. In this talk we discuss two conceptionally different types of such structure conditions. First, we consider weak solutions in the space $W^{1,p}$ in the limiting case $p=n$ (with $n$ the space dimension), where the embedding into the space of continuous functions just fails, and we assume on the inhomogeneity a one-sided condition. Via a double approximation procedure based on variational inequalities, we establish the existence of a weak solution and prove simultaneously its continuity (which, however, does not exclude in general the existence of irregular solutions). Secondly, we consider diagonal systems (with $p=2$) and assume on the inhomogeneity sum coerciveness. Via blow-up techniques we here establish the existence of a regular weak solution and Liouville-type properties. All results presented in this talk are based on joint projects with Jens Frehse (Bonn) and Miroslav Bulíček (Prague).