On criticality theory for elliptic mixed boundary value problems in divergence form

Abstract

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain Ω ⊆ R n that satisfy an oblique boundary condition on a portion of ∂Ω. First, we study the degenerate mixed boundary value problemwhere Ω is a bounded Lipschitz domain, ∂Ω Rob is a relatively open portion of ∂Ω, ∂Ω Dir is a closed set of ∂Ω, and B is an oblique (Robin) boundary operator defined on ∂Ω Rob .In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a positive minimal Green function. Then we establish a criticality theory for positive weak solutions of the operator (P, B) in a general domain Ω with no boundary condition on ∂Ω Dir and no growth condition at infinity. The paper generalizes and extends results obtained by Pinchover and Saadon (2002) for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of (P, B), and the boundary ∂Ω Rob are assumed.

Optimal Hardy-weights for elliptic operators with mixed boundary conditions

“…The following generalized maximum principle for a nonnegative operators (P,B) is proved in [20,Lemma 3.19] and is a consequence of the ground state transform.…”

“…We recall some essential definitions and results which are discussed in detail in [20]. Let P be a second-order, linear, elliptic operator with real measurable coefficients which is defined on a domain Ω ⊂ R n .…”

“…By [20,Theorem 5.19], (P, B) is subcritical in Ω if and only if (P, B) admits a positive minimal Green function G Ω P,B (x, y) in Ω. On the other hand, (P, B) is critical in Ω if and only if the equation (P, B)u = 0 admits (up to a multiplicative constant) a unique positive supersolution φ in Ω.…”

“…In fact, φ is a positive solution, called the (Agmon) ground state (for the definition of a ground state see Definition 2.28). Furthermore, (P, B) is critical in Ω if and only if (P ⋆ , B * ) is critical in Ω, where (P ⋆ , B * ) is the formal adjoint of (P, B) in L 2 (Ω) [20,Corollary 5.20].…”

“…Remark 1.7. For the existence of such an exhaustion see the appendix in [20]. We note that the construction of {Ω k } k∈N ensures that ∂Ω * k,Rob is Lipschitz as well.…”