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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 38 (12 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.
Hardy Inequalities With Optimal Constants and Remainder Terms
, 2003
"... We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higherorder Sobolev spaces on a bounded domain Ω ⊂ Rn can be refined by adding remainder terms which involve Lp norms. In the higherorder case further Lp norms with lowerorder singular weight ..."
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Cited by 30 (0 self)
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We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higherorder Sobolev spaces on a bounded domain Ω ⊂ Rn can be refined by adding remainder terms which involve Lp norms. In the higherorder case further Lp norms with lowerorder singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0.
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 22 (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 15 (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
, 2007
"... 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 11 (3 self)
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In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M. Furthermore, we also obtain sharp constant for the improved
On a class of Rellich inequalities
 Ann. Scuola Norm. Pisa
, 1997
"... We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension k, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of thes ..."
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Cited by 7 (6 self)
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We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension k, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of these inequalities.
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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Cited by 4 (0 self)
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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.
Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators
, 2006
"... We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators Lpu: = − ∇ ∗ L (∇Lu  p−2 ∇Lu). If φ is a positive weight such that −Lpφ ≥ 0, then the Hardy type inequality u ..."
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Cited by 2 (0 self)
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We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators Lpu: = − ∇ ∗ L (∇Lu  p−2 ∇Lu). If φ is a positive weight such that −Lpφ ≥ 0, then the Hardy type inequality u
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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Cited by 1 (0 self)
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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