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36
Explicit Constants for Rellich Inequalities in ...
, 1997
"... Introduction Let\Omega be a bounded region in a complete Riemannian manifold with smooth boundary @ Let C 1 (\Omega ); C 1 0 (\Omega\Gamma and C 1 c (\Omega\Gamma denote respectively the space of smooth functions on\Omega , the subspace consisting of such functions which vanish on @ and the su ..."
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Cited by 15 (2 self)
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Introduction Let\Omega be a bounded region in a complete Riemannian manifold with smooth boundary @ Let C 1 (\Omega ); C 1 0 (\Omega\Gamma and C 1 c (\Omega\Gamma denote respectively the space of smooth functions on\Omega , the subspace consisting of such functions which vanish on @ and the subspace of such functions which vanish in a neighbourhood of @ \Omega\Gamma For those whose main interest is in spectral theory in Euclidean space we mention that our main results are also new in that context. We investigate the existence and explicit determination of constants c and weights X and Y on\Omega such that the Rellich inequality Z \Omega Xjuj p c Z \Omega Y
Series expansion for L p Hardy inequalities
, 2001
"... We consider a general class of sharp Lp Hardy inequalities in IR N involving distance from a surface of general codimension 1 ≤ k ≤ N. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite ser ..."
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Cited by 10 (2 self)
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We consider a general class of sharp Lp Hardy inequalities in IR N involving distance from a surface of general codimension 1 ≤ k ≤ N. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite series improvement of Lp Hardy inequalities. AMS Subject Classification: 35J20 26D10 (46E35 35P)
A review of Hardy inequalities
 Eds.), The Maz'ya Anniversary Collection
, 1999
"... We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation. ..."
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Cited by 9 (0 self)
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We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation.
Perturbative Oscillation Criteria And HardyType Inequalities
"... . We prove a natural generalization of Kneser's oscillation criterion and Hardy's inequality for SturmLiouville differential expressions. In particular, assuming \Gamma d dx p0 (x) d dx + q0 (x), x 2 (a; b), \Gamma1 a ! b 1 to be nonoscillatory near a (or b), we determine conditions on q(x) su ..."
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Cited by 7 (0 self)
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. We prove a natural generalization of Kneser's oscillation criterion and Hardy's inequality for SturmLiouville differential expressions. In particular, assuming \Gamma d dx p0 (x) d dx + q0 (x), x 2 (a; b), \Gamma1 a ! b 1 to be nonoscillatory near a (or b), we determine conditions on q(x) such that \Gamma d dx p 0 (x) d dx + q0 (x) + q(x) is nonoscillatory, respectively, oscillatory near a (or b). 1. Introduction In this note we compare oscillation properties of solutions of SturmLiouville equations ø 0 / 0 = / 0 and ø/ = /, where ø 0 is of the type ø 0 = \Gamma d dx p 0 d dx + q 0 (x) and its perturbation ø is of the form ø = ø 0 + q(x). More precisely, assuming 0 ! p \Gamma1 0 2 L 1 loc (((a; b)); q 0 2 L 1 loc ((a; b)) realvalued; (1.1) consider the (quasi) differential expression ø 0 = \Gamma d dx p 0 (x) d dx + q 0 (x); x 2 (a; b); \Gamma1 a ! b 1 (1.2) and its perturbation ø = ø 0 + q(x); x 2 (a; b); (1.3) where q 2 L 1 loc ((a; b)) is real...
Harmonic metrics and connections with irregular singularities
 Ann. Inst. Fourier (Grenoble
, 1999
"... Abstract. We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. ..."
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Cited by 6 (2 self)
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Abstract. We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L 2 complex. As a consequence, we obtain a Hard Lefschetztype theorem. 1. Statement of the results Let X be a compact Riemann surface, D ⊂ X be a finite set of points and denote by j the ∗ def open inclusion X d, equipped with a connection ∇ : M → M ⊗OX Ω1 X = X − D ֒ → X. Let M be a locally free OX[∗D]module of finite rank which may have regular or irregular singularities at each point of D. Therefore, M is also a holonomic module on the ring DX of holomorphic differential operators on X. We call such a DXmodule a meromorphic connection for short. There exists a unique holonomic DXsubmodule Mmin ⊂ M satisfying (1) OX[∗D] ⊗OX Mmin = M, (2) Mmin has no quotient supported on a subset of D. One says that Mmin is the minimal extension of M along D. If M ∗ denotes the dual DXmodule, we have an exact sequence
Real Interpolation With Logarithmic Functors
, 1996
"... . We present a real interpolation method involving brokenlogarithmic functors. We obtain a variety of interpolation theorems for quasilinear operators on quasiBanach spaces, including limiting cases. We present a number of Holmstedt's formulae corresponding to the brokenlogarithmic functors and a ..."
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Cited by 4 (0 self)
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. We present a real interpolation method involving brokenlogarithmic functors. We obtain a variety of interpolation theorems for quasilinear operators on quasiBanach spaces, including limiting cases. We present a number of Holmstedt's formulae corresponding to the brokenlogarithmic functors and apply these to obtain reiteration theorems for interpolation and extrapolation spaces. We compare the two methods of interpolation and demonstrate the results with examples. 1. Introduction In [EOP], a variety of interpolation theorems for operators of joint weak type (p 0 ; q 0 ; p 1 ; q 1 ) in the sense of Bennett and Rudnick [BR] were proved, using Hardy inequalities and embedding theorems for generalized LorentzZygmund spaces. Although the results cover and improve a lot of known facts and introduce new ones, the method has certain restrictions, especially in the limiting cases of interpolation. Namely, the method is restricted to rather special spaces, and involving functions defined ...
Greedy approximation of highdimensional Ornstein– Uhlenbeck operators with unbounded drift
 Foundations of Computational Mathematics. (In print). Preprint: arXiv:1103.0726v1 [math.NA
"... Abstract. We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (cf. J. NonNewtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of highdimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic m ..."
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Cited by 3 (2 self)
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Abstract. We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (cf. J. NonNewtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of highdimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in R 2, subject to a homogeneous Dirichlet boundary condition, the mathematicalanalysis ofthe algorithmwascarriedoutrecentlyby Le Bris, Lelièvre andMaday (Const. Approx. 30: 621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. In this paper, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris, Lelièvre and Maday to the technically more complicated case where the Laplace operator is replaced by a highdimensional Ornstein–Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker–Planck equations that arise in beadspring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a highdimensional Cartesian product configuration space D = D1 ×···×DN contained in R Nd, where each set Di, i = 1,...,N, is a bounded open ball in R d, d = 2,3. 1.
A Sharp Rearrangement Inequality For Fractional Maximal Operator
, 1999
"... . We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of f , M fl f , by an expression involving the nonincreasing rearrangement of f . This estimate is used to obtain necessary and sufficient conditions for the boundedness of M fl between classi ..."
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Cited by 2 (1 self)
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. We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of f , M fl f , by an expression involving the nonincreasing rearrangement of f . This estimate is used to obtain necessary and sufficient conditions for the boundedness of M fl between classical Lorentz spaces. 1. Introduction and statement of main results For n 2 N and fl 2 [0; n), the fractional maximal operator M fl is defined at f 2 L 1 loc (R n ) by (M fl f)(x) = sup Q3x jQj fl n \Gamma1 Z Q jf(y)j dy; x 2 R n ; where the supremum is extended over all cubes Q ae R n with sides parallel to the coordinate axes and jEj denotes the ndimensional Lebesgue measure of a measurable subset E of R n . For the classical HardyLittlewood maximal operator M := M 0 , the rearrangement inequality (1.1) cf (t) (Mf) (t) Cf (t); t 2 (0; 1); holds, where f (t) = inf n ? 0; jfx 2 R n ; jf(x)j ? gj t o is the nonincreasing rearrangement of f , f (t) = t...
Approximate quantum and acoustic cloaking
, 812
"... Abstract. For any E ≥ 0, we construct a sequence of bounded potentials V E n, n ∈ N, supported in an annular region Bout\Binn ⊂ R 3, which act as approximate cloaks for solutions of Schrödinger’s equation at energy E: For any potential V0 ∈ L ∞ (Binn) such that E is not a Neumann eigenvalue of − ∆ + ..."
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Cited by 1 (1 self)
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Abstract. For any E ≥ 0, we construct a sequence of bounded potentials V E n, n ∈ N, supported in an annular region Bout\Binn ⊂ R 3, which act as approximate cloaks for solutions of Schrödinger’s equation at energy E: For any potential V0 ∈ L ∞ (Binn) such that E is not a Neumann eigenvalue of − ∆ + V0 in Binn, the scattering amplitudes aV0+V E(E, θ, ω) → 0 as n → ∞. The V n E n thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks in quantum mechanics. On the other hand, for E close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in Binn. We derive the V E n from singular, anisotropic transformation opticsbased cloaks by a deanisotropization procedure, which we call isotropic transformation optics. This technique uses truncation, inverse homogenization and spectral theory to produce nonsingular, isotropic approximate cloaks. As an intermediate step, we also obtain approximate cloaking for a general class of equations including the acoustic equation. 1.