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Nonlinear ground state representations and sharp Hardy inequalities
 J. Funct. Anal
"... Abstract. We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a nonlinear and nonlocal version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev e ..."
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Cited by 8 (1 self)
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Abstract. We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a nonlinear and nonlocal version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces. 1. Introduction and
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
A Liouvilletype theorem for Schrödinger operators
, 2005
"... In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator, such that a nonzero solution of another symmetric nonnegative operator is a ground state. In particular, if Pj: = − ∆ + Vj, for j = 0,1, are two nonnegative Schrödinger operato ..."
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Cited by 2 (1 self)
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In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator, such that a nonzero solution of another symmetric nonnegative operator is a ground state. In particular, if Pj: = − ∆ + Vj, for j = 0,1, are two nonnegative Schrödinger operators defined on Ω ⊆ R d such that P1 is critical in Ω with a ground state ϕ, the function ψ ≰ 0 solves the equation P0u = 0 in Ω and satisfies ψ  ≤ Cϕ in Ω, then P0 is critical in Ω and ψ is its ground state. In particular, ψ is (up to a multiplicative constant) the unique positive solution of the equation P0u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.
The Hardy inequality and the heat equation in twisted tubes
, 2009
"... We show that a twist of a threedimensional tube of uniform crosssection yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of selfsimilar variable ..."
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We show that a twist of a threedimensional tube of uniform crosssection yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of selfsimilar variables and weighted Sobolev spaces for the heat equation. Contents 1
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, 901
"... On geometric perturbations of critical Schrödinger operators with a surface interaction ..."
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On geometric perturbations of critical Schrödinger operators with a surface interaction
WEAKLY COUPLED BOUND STATES OF PAULI OPERATORS
, 903
"... Abstract. We consider the twodimensional Pauli operator perturbed by a weakly coupled, attractive potential. We show that besides the eigenvalues arising from the AharonovCasher zero modes there are two or one (depending on whether the flux of the magnetic field is integer or not) additional eigen ..."
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Abstract. We consider the twodimensional Pauli operator perturbed by a weakly coupled, attractive potential. We show that besides the eigenvalues arising from the AharonovCasher zero modes there are two or one (depending on whether the flux of the magnetic field is integer or not) additional eigenvalues for arbitrarily small coupling and we calculate their asymptotics in the weak coupling limit. 1. Introduction and
LARGE TIME BEHAVIOR OF THE HEAT KERNEL OF TWODIMENSIONAL MAGNETIC SCHRÖDINGER OPERATORS
, 2010
"... We study the heat semigroup generated by twodimensional Schrödinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some ondiagonal heat kernel est ..."
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We study the heat semigroup generated by twodimensional Schrödinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some ondiagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the AharonovBohm magnetic field.
Schrödinger operator with a potential term
, 2008
"... Singular Schrödinger equation with external magnetic field admits a representation with a positive Lagrangean density whenever its “nonmagnetic ” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. W ..."
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Singular Schrödinger equation with external magnetic field admits a representation with a positive Lagrangean density whenever its “nonmagnetic ” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L pnorm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field.
EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES
, 2008
"... Abstract. We consider a reaction diffusion equation ut = ∆u+ f(x, u) in RN with initial data in the locally uniform space L̇qU (RN), q ∈ [1,∞), and with dissipative nonlinearities satisfying sf(x, s) ≤ C(x)s2+D(x)s, where C ∈ Lr1U (RN) and 0 ≤ D ∈ Lr2U (RN) for certain r1, r2> N 2. We construc ..."
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Abstract. We consider a reaction diffusion equation ut = ∆u+ f(x, u) in RN with initial data in the locally uniform space L̇qU (RN), q ∈ [1,∞), and with dissipative nonlinearities satisfying sf(x, s) ≤ C(x)s2+D(x)s, where C ∈ Lr1U (RN) and 0 ≤ D ∈ Lr2U (RN) for certain r1, r2> N 2. We construct a global attractor A and show that A is actually contained in an ordered interval [ϕm, ϕM], where ϕm, ϕM ∈ A is a pair of stationary solutions, minimal and maximal respectively, that satisfy ϕm ≤ lim inft→ ∞ u(t;u0) ≤ lim supt→ ∞ u(t;u0) ≤ ϕM uniformly for u0 in bounded subsets of L̇ q U (RN). A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as x  → ∞. In this case the solutions are shown to enter, asymptotically, Lebesque spaces of integrable functions in RN, the attractor attracts in the uniform convergence topology in RN and is a bounded subset of W 2,r(RN) for some r> N/2. Uniqueness of positive solutions is also discussed. 1.