Elliptic and Parabolic Equations - Abstract
Recke, Lutz
The lecture concerns boundary value problems for quasilinear second order elliptic equations in divergence form with nonsmooth data and for weakly coupled systems of such equations. Here nonsmooth data means that the boundary $partial Omega$ of the domain can be nonsmooth (but has to be Lipschitz), that the coefficients of the equations and the boundary conditions may be discontinuous with respect to the space variables (but have to be smooth with respect to the unknown function $u$), and that the boundary conditions can change type (mixed boundary conditions, where the Dirichlet and the Neumann boundary parts can touch along Lipschitz hypersurfaces in $partial Omega$). The coefficients in the equations and the boundary conditions may be local or nonlocal functions of $u$, they can have any growth with respect to $u$, and the space dimension can be arbitrary. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of of quasilinear elliptic PDEs with nonsmooth data.
We apply the Implicit Function Theorem and the Newton Iteration Procedure to those problems. In particular, under natural assumptions we get smooth dependence of the solution on the data and quadratic convergence of the Newton iterations in $W^1,2(Omega) cap C(overlineOmega)$.
The main tools for the proofs are maximal regularity properties of the corresponding linearized problems on Sobolev--Morrey--Campanato spaces of functions and functionals, respectively, and differentiability properties of Nemycki operators.