Abstract :
Suppose that E is a bounded domain of class C2,λ in Rd and L is a uniformly elliptic operator in E. The set U of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions uΓ⩽wΓ. We also introduced a class of solutions uν in 1–1 correspondence with a certain class N0 of σ-finite measures ν on ∂E. With every u∈U we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and ν∈N0. We called this pair the fine boundary trace of u and we denoted in tr(u).
Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class EL,ψ if the condition tr(u)=(Γ,ν) implies that u⩽wΓ⊕uν and we say that u belongs to EL,ψ∗ if the condition tr(u)=(Γ,ν) implies that u⩾uΓ⊕uν.
It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class EL,ψ∗ contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de lʹUniversité Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C4 belong to EΔ,ψ where Δ is the Laplacian and ψ(u)=u2. [Mselati proved that, in his case, uΓ=wΓ and therefore the condition tr(u)=(Γ,ν) implies u=uΓ⊕uν=wΓ⊕uν.]
By modifying Mselatiʹs arguments, we extend his result to ψ(u)=uα with 1<α⩽2 and all bounded domains of class C2,λ.
We start from proving a general localization theorem: E∈EL,ψ under broad assumptions on L, ψ if, for every y∈∂E there exists a domain E′∈EL,ψ such that E′⊂E and ∂E∩∂E′ contains a neighborhood of y in ∂E.
