– last query –
Dr. Kurt Pagani
Remarks on the critical points of solutions to some quasilinear elliptic equations of second order in the plane.
This paper adds new information concerning critical points of a solution u∈C 3 (Ω)∩C 1 (Ω ¯) of a semilinear elliptic equation Δu=f(u,∇u) in a two-dimensional bounded domain Ω, where f∈C 1 and f u ≥0. Under the assumption that the mapping ∇u/|∇u| is injective on ∂Ω, the number of critical points in Ω ¯ is finite and odd, and each critical point is nondegenerate. Especially, in a strictly convex domain Ω, u has exactly one critical point in Ω ¯ provided that u is constant and ∇u never vanishes on ∂Ω. The proof is based on a careful analysis of the nodal lines of directional derivatives of u, which seems peculiar to the two-dimensional case.
Reviewer: K.-I.Nagasaki (Minneapolis)
Cited in 3 Documents