Results 1  10
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13
Hardy Inequalities With Optimal Constants and Remainder Terms
, 2000
"... We show that in the classical Hardy inequalities with optimal constants in W 1;p 0( , W 2;2 0( 8 W 2;2 \W 1;2 0( , W 2;p \W 1;p 0 ( and also in further higher order Sobolev spaces remainder terms may be added. Here is any bounded domain. For the Hardy inequality in W 1;p 0 (1 < p < 1) ..."
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Cited by 18 (1 self)
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We show that in the classical Hardy inequalities with optimal constants in W 1;p 0( , W 2;2 0( 8 W 2;2 \W 1;2 0( , W 2;p \W 1;p 0 ( and also in further higher order Sobolev spaces remainder terms may be added. Here is any bounded domain. For the Hardy inequality in W 1;p 0 (1 < p < 1) a further L p norm appears. The corresponding estimation constant behaves dierently in the cases p 2 and 1 < p < 2. In higher order Sobolev spaces besides the L 2 norm (L p norm resp.) further singularly weighted L 2 norms (L p norms resp.) arise. 1 Introduction Hardy's inequality in dimensions n > 2 8u 2 W 1;2 0( : Z jruj 2 dx (n 2) 2 4 Z u 2 jxj 2 dx (1) is one of the really classical Sobolev embedding inequalities, see [H, HLP]. Here and in what follows, R n is a bounded domain. Although we do not explicitly assume that 0 2 we always have this particularly interesting case in mind. Closely related and proven analogously is the L p versio...
ASYMPTOTICS OF THE FAST DIFFUSION EQUATION VIA ENTROPY ESTIMATES
"... Abstract. We consider nonnegative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction ti ..."
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Cited by 14 (7 self)
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Abstract. We consider nonnegative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a selfsimilar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results. 1.
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
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.
Improved Hardy and Rellich inequalities on Riemannian manifolds, preprint
"... Abstract. In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M. Furthermore, we also obtain sharp constant for the improved ..."
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Cited by 4 (2 self)
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Abstract. In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M. Furthermore, we also obtain sharp constant for the improved
Missing terms in generalized Hardy's inequalities and related topics
, 2002
"... In this article we shall investigate the Hardy inequalities and improve them by nding out missing terms. Although the missing terms for the higher order Hardy inequality can not be determined in a unique way, we shall give a canonical form of the remainder. As a direct application we shall study blo ..."
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In this article we shall investigate the Hardy inequalities and improve them by nding out missing terms. Although the missing terms for the higher order Hardy inequality can not be determined in a unique way, we shall give a canonical form of the remainder. As a direct application we shall study blowup solutions of a semilinear elliptic boundary value problem and give some lower estimate of the rst eigenvalue of the linearized operator. We also improve the weighted Hardy inequalities, which will be fundamental to study singular solutions of quasilinear elliptic equations. 1
The HardyRellich Inequality for . . .
 PROC. ROY. SOC. EDINBURGH SECT. A
, 1999
"... The HardyRellich inequality given here generalizes a Hardy inequality of Davies [2], from the case of the Dirichlet Laplacian of a region\Omega ` R N to that of the higher order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information f ..."
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The HardyRellich inequality given here generalizes a Hardy inequality of Davies [2], from the case of the Dirichlet Laplacian of a region\Omega ` R N to that of the higher order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.
Optimal Sobolev and HardyRellich constants under Navier boundary conditions
"... We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti [19] and an extension argument which enables us to extend radial functions from the ball to the whole space with no increa ..."
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We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti [19] and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for HardyRellich inequalities. AMS Classification: primary 46E35, secondary 26D10, 35J55 Keywords: optimal constant, Sobolev embedding, HardyRellich inequality
Best constants for higherorder Rellich inequalities in L p (Ω)
, 2008
"... We obtain a series improvement to higherorder L pRellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added. ..."
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We obtain a series improvement to higherorder L pRellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added.
Φ(1/x)f(x)  2 ∫
, 2007
"... Abstract. Sharp L p extensions of Pitt’s inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and SteinWeiss potentials. Optimal constants are obtained for the full SteinWeiss potential as a map from L p to itself which in turn yield semiclassical Rellich ..."
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Abstract. Sharp L p extensions of Pitt’s inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and SteinWeiss potentials. Optimal constants are obtained for the full SteinWeiss potential as a map from L p to itself which in turn yield semiclassical Rellich inequalities on R n. Additional results are obtained for SteinWeiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and SteinWeiss estimates. Weighted inequalities for the Fourier transform provide a natural measure to characterize both uncertainty and the balance between functional growth and smoothness. On Rn the question is to determine sharp quantitative comparisons between the relative size of a function and its Fourier transform at infinity. Pitt’s inequality illustrates this principle at the spectral level (see [4], [7]): (1)