Singular problems in elliptic equations and optimal control

The study of Poisson’s equation with general measure data was initiated in the 1920s and has since then been enlarged to more general elliptic equations such as those arising from the Thomas–Fermi model in quantum mechanical theory. The first part of this thesis deals with the existence of weak solutions with zero boundary trace of Schrödinger equations with singular potentials and measure data. For potentials in Lebesgue spaces, we characterize the finite measures for which such solutions exist in terms of first- and second-order Sobolev capacities. The case of potentials that are merely Borel measurable functions is also addressed. The second part of this thesis focus on the boundary behaviour of weak solutions of Schrödinger equations with potentials that are locally summable functions. In particular, we prove that the Hopf boundary point lemma holds except on a set that depends solely on the potential. We also establish a pointwise representation formula for the distributional normal derivative. The third part of this thesis is dedicated to the optimal control of nonlinear elliptic problems with sparsity. Our main result ensures the existence of an optimal control for a wide class of nonlinearities and cost functionals. The Lavrentiev phenomenon is also discussed.(SC - Sciences) -- UCL, 202