Self-adjointe extensions for symmetric Laplacians on polygons.

By the results contained in the paper by Birman and Skvortsov “On the square summability of the highest derivatives of the solution to the Dirichlet problem in a region with piecewise smooth boundary" the Laplace operator o on a plane curvilinear polygon  with domain the Sobolev space H2() and homogeneous Dirichlet boundary conditions is a closed symmetric operator with deficiency indices (n, n), where n is the number of non-convex corners. Therefore on a non-convex polygon, o has infinite self-adjoint extensions. Such extensions have been recently determined by means of Kreǐn's resolvent formula. The purpose of this thesis is to extend such results to the case of different, more general, boundary conditions. In the first part of the thesis we consider the case of mixed Dirichlet-Neumann conditions, thus allowing each side j of the polygon boundary to support either a Dirichlet or a Neumann homogeneous boundary condition. In this case, building on results by Grisvard, while in the pure Neumann case the dimensions of the defect spaces is the same as in the case of the pure Dirichlet case already studied by Birman and Skvortsov, the mixed case has a different behavior, allowing both convex cases and non-convex cases with double vertex contribution. After explicitly characterizing the defect sub- space we determined the self-adjoint extensions by a Kreǐn's resolvent formula proceeding analogously to the pure Dirichlet case, however taking into account the double contribution due to the vertices with mixed boundary conditions. In the second part of the thesis we further extend our analysis by allowing some sides j to support Robin boundary conditions. While this is a deformation of the case considered in the first part, some not completely trivial calculations are necessary in order to get results similar to the ones concerning the mixed Dirichlet-Neumann case. By such calculations, one can recover results anologous to the ones in the first part. However also different behaviors are possible: 1. for any  > 0, for any 0 < j < , there are parameter values which give dj = 1; 2. for any x < j  (3/2) , x  1.43, there are parameter values which give dj = 2. Moreover, as expected, the dj 's converge to the ones corresponding to the mixed Dirichlet-Neumann case as the j 's converge to either 0 or 1 accordingly to the different possible cases and as in the mixed Dirichlet-Neumann a Kreǐn's formula giving the classification of all the self-adjoint extension is provided in Chapter 4.