Prof. Chérif Amrouche “Divergence Elliptic Equations in Lipschitz and in C1 domains”

We are interested here in questions related to the study of some divergence elliptic equations in bounded Lipschitz or C1 domains:
−div(a∇u) + bu = f, in Ω,

with Dirichlet or Neumann boundary condition. We will consider three different cases.

1. We assume a = 1 and b = 0, corresponding to the Laplace equation. We will give some new results on the traces of non smooth functions, harmonic or non-harmonic. Using in particular the interpolation theory, we are going to study the questions of existence and maximal regularity of solutions in fractional Sobolev spaces or with weights associated with the distance to the boundary.
2. We assume that b = 0 and a satisfies the classical condition to ensure the ellipticity of the operator − div(a ∇). We will concentrate on the case of generalized solutions in W^{1,p}(Ω) with 1 < p < ∞.
3. We will finally consider the following problem:

−div (ρ^α∇u) + k u/(ρ^β)= f, in Ω,

with or without boundary condition and where k is a non negative constant and α and β belong to the interval [0, 1].