2016/10/31 by Espinar, José M., Mazet, Laurent
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1610.09833
We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with C2 boundary, i.e., positive solutions u to Δu + f(u)=0 in Ω⊂ (M2,g) so that u = 0 and (∂ u)/(∂ η) = cte along ∂ Ω, η the unit outward normal along ∂Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. When (M2,g) is the standard sphere \mathbb S2 and f is a C1 function so that f(x)>0 and f(x)≥ x f'(x) for any x∈\mathbb R+^*, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki-Caffarelli-Nirenberg conjecture in \mathbb S2 for this choice of f. More precisely, this shows that if u is a positive solution to Δu + f(u) = 0 on a topological disk Ω⊂ \mathbb S2 with C2 boundary so that u = 0 and (∂ u)/(∂ η) = cte along ∂ Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called \it Serrin Problem) in \mathbb S 2.