Hopf's Type Estimates near Singular Boundary Points

Subject of this thesis is the behaviour of the solution of the Laplace-Poisson equation under zero Dirichlet boundary condition near non-regular boundary points. The first part gives an example of a domain where Hopf's Boundary Point Lemma holds true pointwise but not uniformly, therefore the solution operator is not strongly positive. A second result addresses a sharp replacement of Hopf's estimate near the boundary whenever such boundary has a conical point. As a consequence one is able to prove an optimal anti-maximum type result for domains with conical shapes. The last part is concerned with the behaviour of the solution at points where an interface reaches the boundary. At the interface the Poisson equation is not satisfied but instead a jump condition for the normal derivatives appears