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 P (x, y, t) of a general time independent parabolic operator L = ut+P (x, ∂x) which is defined on a noncompact manifoldM . More precisely, we prove that lim t→∞ e0k P (x, y, t) always exists. Here λ0 is the generalized principal eigenvalue of the operator P in M . 2000 Mathematics(More)
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 manifolds, and to point out a number of their consequences. 2000 Mathematics Subject Classification. Primary 35J15; Secondary 35B05, 35C15, 35K10.
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P1, such that a nonzero subsolution of a symmetric nonnegative operator P0 is a ground state. Particularly, if Pj := −∆ + Vj , for j = 0, 1, are two nonnegative Schrödinger operators defined on Ω ⊆ R such that P1 is critical in Ω with a(More)
The paper studies the existence of minimizers for Rayleigh quotients μΩ = 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 . 2000 Mathematics Subject(More)
Let a be a quadratic form associated with a Schrödinger operator L = −∇ · (A∇) + V on a domain Ω ⊂ R. 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 φk ∈ C ∞ 0 (Ω) and a function φ > 0 satisfying Lφ = 0 such that a[φk] → 0, φk → φ locally uniformly in Ω \ {x0}. This(More)
In this paper we study the asymptotic behavior of the ground state energy E(R) of the Schrödinger operator PR = −∆ + V1(x) + V2(x−R), x, R ∈ IR, where the potentials Vi are small perturbations of the Laplacian in IR, n ≥ 3. The methods presented here apply also in the investigation of the ground state energy E(g) of the operator Pg = P + V1(x) + V2(gx), x ∈(More)
Let Ω be a domain in Rd, d ≥ 2, and 1 < p <∞. Fix V ∈ Lloc(Ω). Consider the functional Q and its Gâteaux derivative Q′ given by Q(u) := ∫ Ω (|∇u|+V |u|)dx, 1 p Q(u) := −∇·(|∇u|∇u)+V |u|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 sequence uk ∈ C ∞ 0 (Ω) and 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)