Results 1  10
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12
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
Spectral theory for perturbed Krein Laplacians in nonsmooth domains
, 2010
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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Cited by 7 (7 self)
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients
 J. Analyse Math
"... boundary data in Besov spaces for higher order elliptic systems with rough coefficients ∗ ..."
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Cited by 7 (4 self)
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boundary data in Besov spaces for higher order elliptic systems with rough coefficients ∗
Boundedness of the Hessian of a biharmonic function in a convex domain
, 2006
"... We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions. 1 ..."
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Cited by 3 (0 self)
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We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions. 1
On estimates of biharmonic functions on Lipschitz and convex domains, preprint. ————————————– Svitlana Mayboroda Department of Mathematics, The Ohio State University, 231 W 18th Av
 svitlana@math.ohiostate.edu Vladimir Maz’ya Department of Mathematics, The Ohio State University
"... Abstract. Using Maz’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in R n. For n ≥ 8, combined with a result in [S2], these estimates lead to the solvability of the L p Dirichlet problem for the biharmonic equ ..."
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Abstract. Using Maz’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in R n. For n ≥ 8, combined with a result in [S2], these estimates lead to the solvability of the L p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4. 1.
Boundary Value Problems for Higher Order Parabolic Equations
"... this paper and the treatment of regularity problem are taken directly from Hu's thesis [12]. The extension to the Dirichlet problem is new. The outline of this paper is as follows. Sections 25 study the boundary value problem for the elliptic equation L+iø . The last section, section 6 gives the ad ..."
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this paper and the treatment of regularity problem are taken directly from Hu's thesis [12]. The extension to the Dirichlet problem is new. The outline of this paper is as follows. Sections 25 study the boundary value problem for the elliptic equation L+iø . The last section, section 6 gives the additional arguments needed to obtain results for the parabolic problem. Finally, throughout this paper, we assume that constants may depend on the operator through its ellipticity constant, E, bounds for the coefficients and its order and that constants may depend on the Lipschitz constant for the function whose graph defines the domain. Also, constants may depend on the parameter fi defining the approach region. All other dependencies will be denoted explicitly. 2 Estimates for solutions of (L + i)u = 0. Throughout this section, we assume that\Omega is a Lipschitz graph domain. The operator
Spectral Theory for . . . IN NONSMOOTH DOMAINS
, 2009
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
, 2012
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed ..."
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω;d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u,
A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed ..."
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L2 (Ω; dnx) for Ω ⊂ Rn an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0 (Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S.
ON THE BIHARMONIC DIRICHLET PROBLEM: THE HIGHER DIMENSIONAL CASE
, 2001
"... Abstract. We address the question for existence and uniqueness for the biharmonic equation on Lipschitz domains. In particular for the Dirichlet biharmonic problem on D ⊂ R n, we show solvability for data in L p, 2 − < p < 2(n − 1)/(n − 3)+. This result complements known counterexamples due to Mazy’ ..."
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Abstract. We address the question for existence and uniqueness for the biharmonic equation on Lipschitz domains. In particular for the Dirichlet biharmonic problem on D ⊂ R n, we show solvability for data in L p, 2 − < p < 2(n − 1)/(n − 3)+. This result complements known counterexamples due to Mazy’aNazarovPlamenevskii and VerchotaPipher, and is thus sharp at least in dimensions four and five. 1.