Characterization of f-extremal disks

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 [19] to the realm of overdetermined elliptic problem. When (M2,g) is the standard sphere S2 and f is a C1 function so that f(x)>0 and f(x)≥xf′(x) for any x∈R+⁎, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in 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⊂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 (cf. [33,35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in S2.