This paper is concerned with the state-constrained optimal control of the twodimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabo...

In this study we consider the recovery of smooth region boundaries of piecewise constant coefficients of an elliptic PDE −∇ · a∇Φ + bΦ = f from data on the exterior boundary ∂Ω. The assumption is that the values of the coefficients (a, b) are known a priori but the information about the geometry of the smooth region boundaries where a and b are discontinous is missing. For the full characterisa...

We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside of the unit square D, with overposed boundary data. This problem is not investigated enough in the literature due to lack of results about the uniqueness of the inverse problem. Following the main idea of the so-called Method of Variational Imbedding (MVI), we embed the solu...

We establish an extension of Liouville’s classical representation theorem for solutions of the partial differential equation ∆u = 4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {zj} in the unit disk there is always a Blasc...