Abstract: In this talk we will discuss some rigidity results for bounded positive solutions of the general capillary overdetermined problem: div(∇u√1+|∇u|2)+f(u)=0div(∇u1+|∇u|2)+f(u)=0 in ΩΩ, u=0u=0 on ∂Ω∂Ω, ∂νu=const∂νu=const on ∂Ω∂Ω. Our main theorem states that in dimension 2, under some natural assumptions on the function f and the boundedness of |∇u||∇u|, the existence of a solution of the previous problem in a domain di eomorphic to a half-space implies that is a half-space and u is a one dimensional function. We also prove the boundedness of the gradient of the solution when f′(u)<0f′(u)<0. With this, our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where f is linear. This is a joint work with Y. Lian.
Per informazioni, rivolgersi a: galise@mat.uniroma1.it