| The main body of this dissertation is concerned with some linear and semilinear elliptic equations, all of them in a bounded smooth domain of . In part of this work we deal with semilinear elliptic equations involving a positive parameter. The existence of solutions depends very strongly on this parameter, typically with an extremal value beyond which no solutions exist. This is called the extremal parameter, and associated with it there is an extremal solution. First we study qualitative properties of this extremal solution in a particular class of such problems. We also investigate the asymptotic behavior of the extremal parameter in a family of problems, where the boundary condition is approaching a limit configuration in some specific sense. Next, we obtain some comparison results for linear elliptic and parabolic equations. On one hand we prove a result for the Laplace equation with mixed boundary condition, that is, with zero Dirichlet data on part of the boundary and zero Neumann data on the rest. This result is analogous, in some sense, to the well known Hopf boundary lemma, a difference being that here the barrier function is not given explicitly, but as the solution to an appropriate equation. We then consider some linear operators with a singular coefficient, for which the standard regularity theory does not apply. We derive comparison principles for solutions and eigenfunctions of these operators, which typically exhibit singularities. Finally, we address a question on functions of bounded variation. We prove that for such functions, the total mass of the gradient can be expressed as a limit of a double integral, which is related to the characterization of fractional Sobolev spaces. |
