14.30-15.30 : Federica Raimondi (Paris)
A class of singular elliptic problems
Abstract: We investigate a class of quasilinear elliptic problems posed in a domain perforated by epsilon-periodic holes of epsilon-size. The quasilinear equation presents a nonlinear singular lower order term, which is the product of a continuous function z (singular in zero) and a function f whose summability depends on the growth of z near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes and a homogeneous Dirichlet condition on the exterior boundary. The main tool for proving the homogenization result is a convergence result stating that the gradient of our solution behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence as epsilon goes to zero. This idea was originally introduced in the literature for the homogenization of nonlinear problems with quadratic growth with respect to the gradient. In our case, this allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also obtain a corrector result for our problem.
15.30-16.00: Coffee break
16.00-17.00 : Thomas Schmidt (Hamburg), Duality theory for (thin) obstacle problems in BV
Abstract: The talk is concerned with obstacle problems for the total variation and the area integral on the space of functions of bounded variation. It is planned to discuss products of L∞ functions and measures and to present a relaxation and convex duality theory, which explains the precise connection to (super)solutions for the 1-Laplace and minimal surface PDEs. Notably, the theory covers very general obstacles, e.g. thin obstacles and up-to-the-boundary obstacles. The talk is based on joint works with C. Scheven (Duisburg-Essen).
Heiner Olbermann
