Bazı aşırı belirğin problemlerde simetri

In the present study, overdetermined symmetry problems for elliptic and parabolic cases are studied for one-phase, two-phase and multi-phase versions. The study contains five main sections. In the first section, a general introduction is given including some information about symmetry in elliptic and parabolic multi-phase overdetermined problems with nonlinear governing equations. Existing results in literature are also argued briefly. The second section is devolved to the background. Related terminology with basic definitions and theorems about Laplace and heat equations are reviewed. In the third section, symmetry for an elliptic overdetermined problem, related to balls is analysed. A general overview of existing results is given. First, Laplace equation with nonlinear boundary conditions is considered for both one-phase and two-phase cases. For one-phase case the nature of the problem is indicated. The difficulties of the problem is observed for the two-phase case. Then, p-Laplacian operator is considered and in order to generalize the results, exact definitions and forms of the Green's functions for the ball are written. Multi-phase version of the problem for the p-Laplacian is examined in detail. These results also cover the symmetry for the Laplace equation for multi-phases. In the fourth section, symmetry for a parabolic overdetermined problem, related to heat balls is analysed. The one-phase, two-phase and multi-phase versions with nonlinear overdetermined boundary conditions are proved. Along with these ideas spherical space symmetry problems for the solutions of similar overdetermined problems are also stated and proved.