Universitat Ausburg

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).