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15
Concentrationcompactness phenomena in the higher order Liouville’s equation
"... We investigate different concentrationcompactness phenomena related to the Qcurvature in arbitrary even dimension. We first treat the case of an open domain in R 2m, then that of a closed manifold and, finally, the particular case of the sphere S 2m. In all cases we allow the sign of the Qcurvatu ..."
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Cited by 7 (7 self)
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We investigate different concentrationcompactness phenomena related to the Qcurvature in arbitrary even dimension. We first treat the case of an open domain in R 2m, then that of a closed manifold and, finally, the particular case of the sphere S 2m. In all cases we allow the sign of the Qcurvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R 2m, concentration phenomena can occur only at points of positive Qcurvature. As a consequence, on a locally conformally flat manifold of nonpositive Euler characteristic we always have compactness. 1 Introduction and statement of the main results Before stating our results, we recall a few facts concerning the Paneitz operator P 2m g and the Qcurvature Q2m g on a 2mdimensional smooth Riemannian manifold (M, g). Introduced in [BO], [Pan], [Bra] and [GJMS], the Paneitz operator
Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition
 Indiana Univ. Math. J
"... Abstract. We consider asymptotic behavior of the following fourth order equation ∆ 2 e u = ρ u R Ω eu in Ω, u = ∂νu = 0 on ∂Ω dx where Ω is a smooth oriented bounded domain in R4. Assuming that 0 < ρ ≤ C, we completely characterize the asymptotic behavior of the unbounded solutions. 1. ..."
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Cited by 5 (1 self)
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Abstract. We consider asymptotic behavior of the following fourth order equation ∆ 2 e u = ρ u R Ω eu in Ω, u = ∂νu = 0 on ∂Ω dx where Ω is a smooth oriented bounded domain in R4. Assuming that 0 < ρ ≤ C, we completely characterize the asymptotic behavior of the unbounded solutions. 1.
A threshold phenomenon for embeddings of H m 0 into Orlicz spaces
, 2009
"... Given an open bounded domain Ω ⊂ R 2m with smooth boundary, we consider a sequence (uk)k∈N of positive smooth solutions to j m (−∆) uk = λkuke mu2 k in Ω uk = ∂νuk =... = ∂ m−1 ν uk = 0 on ∂Ω, where λk → 0 +. Assuming that the sequence is bounded in H m 0 (Ω), we study its blowup behavior. We show ..."
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Cited by 3 (3 self)
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Given an open bounded domain Ω ⊂ R 2m with smooth boundary, we consider a sequence (uk)k∈N of positive smooth solutions to j m (−∆) uk = λkuke mu2 k in Ω uk = ∂νuk =... = ∂ m−1 ν uk = 0 on ∂Ω, where λk → 0 +. Assuming that the sequence is bounded in H m 0 (Ω), we study its blowup behavior. We show that if the sequence is not precompact, then lim inf k→ ∞ ‖uk ‖ 2 H m 0 Z: = lim inf uk(−∆) k→∞
Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions, submitted
"... Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equ ..."
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Cited by 2 (0 self)
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Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n, the negative part of the corresponding Green’s function is “small ” when compared with its singular positive part, provided n ≥ 3. Moreover, the biharmonic Green’s function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3. Keywords: Biharmonic Green’s functions, positivity, almost positivity, blowup procedure.
Separating positivity and regularity for fourth order Dirichlet problems in 2ddomains
"... The main result in this paper is that the solution operator for the bilaplace problem with zero Dirichlet boundary conditions on a bounded smooth 2ddomain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part conta ..."
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The main result in this paper is that the solution operator for the bilaplace problem with zero Dirichlet boundary conditions on a bounded smooth 2ddomain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fillup of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bilaplacian on each of these domains is strictly positive.
POSITIVITY ISSUES OF BIHARMONIC GREEN’S FUNCTIONS UNDER DIRICHLET BOUNDARY CONDITIONS
"... Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equ ..."
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Cited by 1 (0 self)
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Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n, the negative part of the corresponding Green’s function is “small ” when compared with its singular positive part, provided n ≥ 3. Moreover, the biharmonic Green’s function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3. 1.
where Ω0
, 2010
"... We discuss compactness, blowup and quantization phenomena for the prescribed Qcurvature equation (−∆) m uk = Vke 2muk on open domains of R 2m. Under natural integral assumptions we show that when blowup occurs, up to a subsequence lim Vke ..."
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We discuss compactness, blowup and quantization phenomena for the prescribed Qcurvature equation (−∆) m uk = Vke 2muk on open domains of R 2m. Under natural integral assumptions we show that when blowup occurs, up to a subsequence lim Vke
OPTIMAL ESTIMATES FROM BELOW FOR BIHARMONIC GREEN FUNCTIONS
"... Abstract. Optimal pointwise estimates are derived for the biharmonic Green function in arbitrary C 4,γsmooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and h ..."
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Abstract. Optimal pointwise estimates are derived for the biharmonic Green function in arbitrary C 4,γsmooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary. 1.