Time: September 9, 2026 14:00
S2 416-1

Titel: Elliptic Equations on Riemannian Manifolds

Abstract: In this talk, we study elliptic equations on compact Riemannian manifolds, with a particular focus on non-linear problems involving the Laplace–Beltrami operator. We first introduce the analytical framework of Sobolev spaces and weak solutions, and discuss the role of Sobolev embeddings and compactness. We then consider the non-linear equation
Δgu + au = uq,
and explain how a variational argument gives the existence of solutions in the subcritical case. Finally, we discuss the critical case, where the loss of compactness of the Sobolev embedding creates the main difficulty, and motivate the approximation of the critical problem by subcritical ones.