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Stable solutions for the bilaplacian with exponential nonlinearity
 SIAM J. Math. Anal
"... Abstract. Let λ ∗> 0 denote the largest possible value of λ such that 8 < ∆ 2 u = λe u ..."
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Abstract. Let λ ∗> 0 denote the largest possible value of λ such that 8 < ∆ 2 u = λe u
Essential selfadjointness of Schrödinger type operators on manifolds
 RUSS. MATH. SURVEYS
, 2002
"... We obtain several essential selfadjointness conditions for the Schrödinger type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bu ..."
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Cited by 4 (1 self)
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We obtain several essential selfadjointness conditions for the Schrödinger type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV. These results generalize the
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.
Fakultät für Mathematik
, 2000
"... 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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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. 1
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
Let
"... An analysis is given of the spectral properties of perturbations of the magnetic biharmonic operator ∆ 2 A in L 2 (R n), n=2,3,4, where A is a magnetic vector potential of AharonovBohm type, and bounds for the number of negative eigenvalues are established. Key elements of the proofs are newly der ..."
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An analysis is given of the spectral properties of perturbations of the magnetic biharmonic operator ∆ 2 A in L 2 (R n), n=2,3,4, where A is a magnetic vector potential of AharonovBohm type, and bounds for the number of negative eigenvalues are established. Key elements of the proofs are newly derived Rellich inequalities for ∆ 2 A which are shown to have a bearing on the limiting cases of embedding theorems for Sobolev spaces H 2 (R n). Copyright line will be provided by the publisher