Yehuda Pinchover

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Let Ω be a domain in Rn and p ∈ (1,∞). We consider the (generalized) Hardy inequality ∫ Ω |∇u|p ≥ K ∫ Ω |u/δ|p, where δ(x) = dist (x, ∂Ω). The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant μp(Ω) = inf ◦ W 1,p(Ω) (∫ Ω |∇u|p / ∫ Ω(More)
In this paper we study the large time behavior of the (minimal) heat kernel k M P (x, y, t) of a general time independent parabolic operator L = u t +P (x, ∂ x) which is defined on a noncompact manifold M. More precisely, we prove that lim t→∞ e λ0t k M P (x, y, t) always exists. Here λ 0 is the generalized principal eigenvalue of the operator P in M .
Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations Abstract The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian man-ifolds, and to point out a number(More)
Let a be a quadratic form associated with a Schrödinger operator L = −∇ · (A∇) + V on a domain Ω ⊂ R d. If a is nonnegative on C ∞ 0 (Ω), then either there is W > 0 such that W |u| 2 dx ≤ a[u] for all C ∞ 0 (Ω; R), or there is a sequence ϕ k ∈ C ∞ 0 (Ω) and a function ϕ > 0 satisfying Lϕ = 0 such that a[ϕ k ] → 0, ϕ k → ϕ locally uniformly in Ω \ {x 0 }.(More)
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P 1 , such that a nonzero subsolution of a symmetric nonnegative operator P 0 is a ground state. Particularly, if P j := −∆ + V j , for j = 0, 1, are two nonnegative Schrödinger operators defined on Ω ⊆ R d such that P 1 is critical in(More)
Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given by Q(u) := Ω (|∇u| p +V |u| p)dx, 1 p Q ′ (u) := −∇·(|∇u| p−2 ∇u)+V |u| p−2 u. If Q ≥ 0 on C ∞ 0 (Ω), then either there is a positive continuous function W such that W |u| p dx ≤ Q(u) for all u ∈ C ∞ 0 (Ω), or there is a(More)
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given(More)
The paper studies the existence of minimizers for Rayleigh quotients µ Ω = inf Ω |∇u| 2 Ω V |u| 2 , where Ω is a domain in R N , and V is a nonzero nonnegative function that may have singularities on ∂Ω. As a model for our results one can take Ω to be a Lipschitz cone and V to be the Hardy potential V (x) = 1 |x| 2 .