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**11 - 18**of**18**### The Hardy-Rellich Inequality for . . .

- PROC. ROY. SOC. EDINBURGH SECT. A
, 1999

"... The Hardy-Rellich 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 Hardy-Rellich 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.

### HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS

, 704

"... Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields: ∇γ = (∇x, |x | 2γ ∇y) where γ> 0, ∇x and ∇y are usual gradient operators in the variables x ∈ R m and y ∈ R k, respectively. In the first part of the paper, we prove some weighted Hardy type i ..."

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Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields: ∇γ = (∇x, |x | 2γ ∇y) where γ> 0, ∇x and ∇y are usual gradient operators in the variables x ∈ R m and y ∈ R k, respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains. 1.

### 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 blow-up 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

### Optimal Sobolev and Hardy-Rellich 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 Hardy-Rellich inequalities. AMS Classification: primary 46E35, secondary 26D10, 35J55 Keywords: optimal constant, Sobolev embedding, Hardy-Rellich inequality

### unknown title

"... Abstract. We systematically study weighted Poincare type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics res ..."

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Abstract. We systematically study weighted Poincare type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics results for fast diusion equations. Inegalites de Hardy-Poincare et applications. Resume. Nous etudions des inegalites de Poincare qui sont etroitement reliees a des inegalites de type Hardy et etablissons la forme des constantes optimales dans certains cas. De telles inegalites sont ensuite utilisees pour relier l'entropie avec la production d'entropie et obtenir des resultats d'asymptotiques intermediaires pour les equations a diusion rapide. Version francaise abregee Dans cette note, nous nous interessons a l'inegalite de Hardy-Poincare

### Hans-Christoph Grunau† Fachgruppe Mathematik

, 2000

"... We show that in the classical Hardy inequalities with optimal constants inW 1,p0 (Ω),W 2,2 0 (Ω), W 2,2∩W 1,20 (Ω) and also in further higher order Sobolev spaces remainder terms may be added. Here Ω is any bounded domain. For the Hardy inequality in W 1,p0 there is a further L p-norm provided p ≥ 2 ..."

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We show that in the classical Hardy inequalities with optimal constants inW 1,p0 (Ω),W 2,2 0 (Ω), W 2,2∩W 1,20 (Ω) and also in further higher order Sobolev spaces remainder terms may be added. Here Ω is any bounded domain. For the Hardy inequality in W 1,p0 there is a further L p-norm provided p ≥ 2, while for 1 < p < 2 we obtain a remainder term in Lq-norms with q < p. In higher order Sobolev spaces besides the L2-norm further singularly weighted L2-norms arise.