Sumários
Introduction to Non-Linear problems, the Obstacle Problem: part IV
30 Maio 2017, 14:00 • Hugo Tavares
The obstacle problem for the minimal surface operator; Lipschitz solutions of the unilateral Plateau problem for surfaces of constant mean curvature over a Lipschitz obstacle in a bounded domain with boundary with compatible mean curvature — the a priori estimate on the global bounded of the gradient (idea of the proof, only); extension to the case of merely continuous obstacles by the Perron method — the existence and uniqueness of a UAG-solution (Uniformly Approximated Generalized solution) to the obstacle problem yielding a mere continuous surface of generalized mean curvature over the obstacle.
Introduction to Non-Linear problems, the Obstacle Problem: part III
25 Maio 2017, 11:00 • Hugo Tavares
Bounded penalization of the obstacle problem for a linear elliptic operator of second order and monotone, uniform and energy approximations of the solution. The Lewy-Stampacchia inequality and its consequences to the regularity of the solution to the obstacle problem according to the smoothness of the data.
Introduction to Non-Linear problems, the Obstacle Problem: part II
23 Maio 2017, 14:00 • Hugo Tavares
Comparison of solutions and maximum principles for strictly T-monotone operators on convex sets; application to the abstract obstacle problem in Hilbertian Vector Lattices; properties of supersolutions of the obstacle problem and the minimality of its solution, application to L^infty estimates in the concrete obstacle problem for Laplacian type operators.
Introduction to Non-Linear problems, the Obstacle Problem: part I
18 Maio 2017, 11:00 • Hugo Tavares
Introduction to the obstacle problem as an elastic and a geometrical equilibrium problems; the variational inequality formulation and the interpretation as a free boundary problem. The well-posedness of variational solutions as the projection over a convex set in the Sobolev space H^1_o in the case of linear elliptic operators of second order with Dirichlet boundary condition. Extension to the non symmetric linear and monotone operators — the Lions-Stampacchia theorem. The case of the minimal surfaces quasilinear operator and the need of a priori estimates on the gradient.
Evolution Problems: part IV
16 Maio 2017, 14:00 • Hugo Tavares
Applications of the semigroup theory to the heat and wave equation. The Dirichlet Laplacian operator.