Open Access

Sidy Moctar Djitte

• The thesis is composed of four Chapters. In the first Chapter, the boundary expression of the one-sided shape derivative of nonlocal Sobolev best constants is derived. As a simple consequence, we obtain the fractional version of the so-called Hadamard formula for the torsional rigidity and the first Dirichlet eigenvalue. An application to the optimal obstacle placement problem for the torsional rigidity and the first eigenvalue of the fractional Laplacian is given. In the second Chapter, we introduce and prove a new maximum principle for doubly antisymmetric functions. The latter can be seen as the first step towards studying the optimal obstacle placement problem for the second fractional eigenvalue. Using the new maximum principle we derive new symmetry results for odd solutions to semilinear Dirichlet boundary value problems with Lipschitz nonlinearity. In the third Chapter, we derive new integration by parts formula for the fractional Laplace operator with a general globally Lipschitz vector field and in particular, we obtain a new Pohozaev type identity generalizing the one obtained by X. Ros-Oton and J. Serra. As an application we obtain nonexistence results for semilinear Dirichlet boundary problems in bounded domains that are not necessarly starshaped. In the last Chapter, we study symmetry properties of second eigenfunctions of annuli. Using results from the first Chapter and the maximum principle in Chpater 2, we extend the result on the optimal obstacle placement problem from the first eigenvalue to the second eigenvalue.