Speaker
Description
In the talk we plan to discuss Dirichlet problem for non-strongly elliptic second-order PDE with constant complex coefficients in bounded simply connected domains in the complex plane. Starting with the most known case of bianalytic functions (that corresponds to the Bitsadze equation), we will proceed to discuss the case of solutions to general non-strongly elliptic second order PDEs with constant complex coefficients and, moreover, solutions to general non-strongly elliptic systems of second-order PDEs with constant coefficients. We will show that any Jordan domain in the complex plane with sufficiently regular (smooth) boundary is not regular with respect to the Dirichlet problem for any non-strongly elliptic system under consideration, which means that there always exists a continuous complex-valued function on the boundary of the domain under consideration that can not be continuously extended to this domain to a function satisfying the corresponding system therein. Since there exists a Jordan domain with Lipschitz boundary, which is regular with respect to the Dirichlet problem for bianalytic functions, the result obtained is near to be sharp. The discovered phenomena that domains with sufficiently smooth boundaries are not regular with respect to the Dirichlet problem for systems under consideration, while domains having worse boundaries may be regular is rather unexpected and essentially new.
