Critical points of solutions to the obstacle problem in the plane
On the unit disk we study the Moser-Trudinger functional and its restrictions , where for . We prove that if a sequence of positive critical points of (for some ) blows up as , then , and weakly in and strongly in . Using this fact we also prove that when is large enough, then has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.
Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak...
Consider the family uₜ = Δu + G(u), t > 0, , , t > 0, , of semilinear Neumann boundary value problems, where, for ε > 0 small, the set is a thin domain in , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of . If G is dissipative, then equation has a global attractor . We identify a “limit” equation for the family , prove convergence of trajectories and establish an upper semicontinuity result for the family as ε → 0⁺.
For a principal type pseudodifferential operator, we prove that condition implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from for any (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition doesnotimply local solvability with a loss of 1 derivative,...