## L p spectral theory of higher-order elliptic differential operators (1997)

Venue: | Bull. London Math. Soc |

Citations: | 15 - 1 self |

### BibTeX

@ARTICLE{Davies97lp,

author = {E. B. Davies},

title = {L p spectral theory of higher-order elliptic differential operators},

journal = {Bull. London Math. Soc},

year = {1997},

volume = {29},

pages = {513--546}

}

### Years of Citing Articles

### OpenURL

### Abstract

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

### Citations

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Stegun (Eds.), Handbook of mathematical functions: With formulas, graphs, and mathematical tables
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Citation Context ... mathematical interest, I should mention the problem of the ideal column, ‘solved’ incorrectly by Lagrange and others before its eventual resolution in [45]. If a is a positive continuous function on =-=[0,1]-=-, then the buckling load for a column of variable cross-sectional area a and length 1 is, after normalization, obtained by finding the smallest eigenvalue of the differential equation (a(x)�f�(x))��λf... |

2174 |
Elliptic partial differential equations of second order, 3rd edition
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Citation Context ...depends upon recasting them as linear equations whose coefficients depend upon the solution and using the properties of the linear equations under the weakest possible assumptions on the coefficients =-=[84, 138]-=-. Finally, we mention that Chapter 3 of Kenig [106] contains a long list of open problems closely related to the subject matter of this review. The present article contains the solution of a small num... |

2077 |
Perturbation Theory for Linear Operators
- Kato
- 1966
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Citation Context ...c�0 and all ξ�CN. This forces the quadratic form Q( f )�� � RN N a (x) i,j i,j=� �f �f� dx (7.1.3) �x �x j i defined on W�,�(RN) to be sectorial, and the associated operator H to be maximal accretive =-=[105, 125]-=-. Every such operator has a sectorial angle α(H) defined as the supremum of the angles θ such that e�iθH still lies in the stated class. If H is selfadjoint, then the sectorial angle is π�2, but in ge... |

1892 | Sobolev Spaces - Adams - 1975 |

1379 | Singular Integrals and Differentiability Properties of Functions - Stein - 1970 |

548 |
Asymptotic Analysis for Periodic Structures
- Bensoussan, Lions, et al.
- 1978
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Citation Context ...rator Hγ has coefficients which are periodic with period γ, then there often exists an operator H with constant coefficients obtained as the γ�0 limit of Hγ. This subject is studied systematically in =-=[145]-=-, and the one-dimensional case is explored in more detail in [51]. Suffice it to say that the coefficients of the limit operator are not simply the averages of the coefficients of the periodic approxi... |

369 |
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions
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Citation Context ...ghest-order coefficients, for example uniform or Ho� lder continuity, then the situation for higher-order elliptic operators in Lp is well understood. Unfortunately, the formulation of the results in =-=[3, 4, 83, 138]-=- involves so many definitions that we must leave the interested reader to consult those works. Henceforth we consider only elliptic operators of order 2m�4 with bounded measurable coefficients. Consid... |

299 |
Trace ideals and their applications
- Simon
- 1979
(Show Context)
Citation Context ...� (ξ)dξ. (2.3.5) If N�2m, then by suitable choices of f,g in the above formula, one may prove that there exists c such that � �V��f��dx�c� �(�∆)mf �f�dx (2.3.6) RN RN for all f�Wm,�, provided V�LN/�m =-=[161, 33]-=-. There are several ways of characterizing weak the non-negative measures µ for which � �f��dµ�c� �(�∆)mf �f�dx (2.3.7) RN RN holds for some constant c and all f�Wm,� [127, Theorems 2.5.2, 12.5.1; 132... |

240 |
Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext
- Ma, Röckner
- 1992
(Show Context)
Citation Context ...c�0 and all ξ�CN. This forces the quadratic form Q( f )�� � RN N a (x) i,j i,j=� �f �f� dx (7.1.3) �x �x j i defined on W�,�(RN) to be sectorial, and the associated operator H to be maximal accretive =-=[105, 125]-=-. Every such operator has a sectorial angle α(H) defined as the supremum of the angles θ such that e�iθH still lies in the stated class. If H is selfadjoint, then the sectorial angle is π�2, but in ge... |

200 |
Methods of mathematical physics
- Courant, Hilbert
- 1953
(Show Context)
Citation Context ... information than that available from the heat equation. There is a long history of improvements of the above results, providing sharper estimates of the remainder for more general elliptic operators =-=[35, 42, 135, 147]-=-. For a general elliptic operator with smooth coefficients acting in a region (or C � manifold) with smooth boundary, two-term spectral asymptotics of the form N(λ)�aλN/�m�bλ(N−�)/�m�o(λ(N−�)/�m) (2.1... |

185 | Sobolev Spaces - Maz’ya - 1985 |

163 |
Heat kernels and spectral theory. Cambridge Tracts
- Davies
- 1989
(Show Context)
Citation Context .... IfHis second-order elliptic acting in L�(RN), then there are Gaussian lower bounds on the heat kernel which complement the upper bounds, provided that the coefficients of the operator are real; see =-=[168, 50, 152]-=- for the extensive literature on this problem, and its generalization to Riemannian manifolds M under suitable geometrical conditions on the latter. If M is noncompact, then a principal object of inte... |

137 | One-Parameter Semigroups - Davies - 1980 |

133 |
and Quasilinear Parabolic Problems
- Amann, Linear
- 1995
(Show Context)
Citation Context ...complex powers Az and Bz are sufficiently wellbehaved. It turns out that the theory depends upon � being a ζ-convex space, in the sense that the Hilbert transform is bounded on Lp(R,�) for 1�p��. See =-=[6, 65]-=- for more details concerning the proper interpretation of the above formula and its applications. We conclude by mentioning that other aspects of the theory of the H � functional calculus also depend ... |

127 |
Functional Integration and Quantum Physics
- Simon
- 1979
(Show Context)
Citation Context ... on Lp for all p�[1,�) [50, Theorem 1.3.5]. The integral kernel of e−Ht is given by the famous Feynman–Kac formula, and the whole theory is profoundly enriched by its connections with Brownian motion =-=[162, 163]-=-. The situation for 2m�2 is much more complicated, and depends upon the dimension and order. If N�2m, then the Schro� dinger semigroup is strongly continuous and uniformly bounded on Lp for all 1�p��.... |

107 |
Harmonic analysis techniques for second order elliptic boundary value problems
- Kenig
(Show Context)
Citation Context ...oefficients depend upon the solution and using the properties of the linear equations under the weakest possible assumptions on the coefficients [84, 138]. Finally, we mention that Chapter 3 of Kenig =-=[106]-=- contains a long list of open problems closely related to the subject matter of this review. The present article contains the solution of a small number of these, but much remains to be done! With a v... |

89 |
An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Norm
- Meyers
- 1963
(Show Context)
Citation Context ...n in [29, 30, 32]. The approach of Barbatis [18, 19, 20] concentrates on the comparison of small eigenvalues and the corresponding resolvent operators. It depends upon a fundamental theorem of Meyers =-=[136]-=- to the effect that if 2m�2, then there exists p�2 such that (H�1)−�Lp�W�,p, and a higher-order analogue of this result in [15, 20]. Barbatis [20] also obtained estimates of the type �(H �1)−��(H�1)−�... |

81 |
The heat equation on non-compact Riemannian manifolds
- Grigor’yan
- 1991
(Show Context)
Citation Context ...uation is, however, much simpler for manifolds with non-negative Ricci curvature, and for the manifolds with the volumedoubling property vol(x,2r)�cvol(x,r) (5.5.2) for some c�0 and all x�M, r�0; see =-=[87, 152, 156, 168]-=- for references to some of the enormous recent literature on this problem. We draw particular attention to [40], where diagonal lower bounds on K(t,x,x) are obtained without recourse to the use of the... |

61 |
A note on Poincaré, Sobolev, and Harnack inequalities
- Saloff-Coste
- 1992
(Show Context)
Citation Context ...uation is, however, much simpler for manifolds with non-negative Ricci curvature, and for the manifolds with the volumedoubling property vol(x,2r)�cvol(x,r) (5.5.2) for some c�0 and all x�M, r�0; see =-=[87, 152, 156, 168]-=- for references to some of the enormous recent literature on this problem. We draw particular attention to [40], where diagonal lower bounds on K(t,x,x) are obtained without recourse to the use of the... |

58 |
Boundary-value problems for elliptic equations in domains with conical or angular points
- Kondrat’ev
- 1967
(Show Context)
Citation Context ...xamples in the subject depend upon the asymptotic expansion of solutions of elliptic equations at a conical point, a subject initiated by Kondrat’ev, Maz’ya, Plamenevskii and others in the 1960s; see =-=[107, 130]-=-. Let Ω be a region in the unit sphere SN−� of RN which has a non-empty smooth boundary, and define its associated conical region with vertex at the origin by C Ω,r ��x:0��x��r,x��x��Ω�. (3.6.1) Let H... |

57 |
Über eine von Rayleigh formulierte minimaleigenschaft des kreises
- Krahn
- 1925
(Show Context)
Citation Context ...ect to Dirichlet boundary conditions has minimum principal eigenvalue is a circular disc, and is known as the isoperimetric problem for a membrane. The higher-dimensional analogue was solved by Krahn =-=[115, 116]-=- in 1925�26. The second conjecture was that among all clamped plates with a given area, the circular disc again has the minimal principal eigenvalue λ � (Ω). The clamped plate may be identified mathem... |

52 |
Viscous and Resistive Eddies Near a Sharp Corner
- Moffatt
- 1964
(Show Context)
Citation Context ...tudy of stresses at inwardly pointing cracks in solid mechanics is clearly not going to be helped by smoothing out the crack [113]. The analysis of ‘Moffatt’ vortices of incompressible viscous fluids =-=[134, 137]-=- depends upon some rather interesting properties of the biharmonic equation at corners. Finally, a substantial part of the theory of non-linear PDEs depends upon recasting them as linear equations who... |

51 |
Microlocal dispersive smoothing for the Schrödinger equation
- Craig, Kappeler, et al.
- 1996
(Show Context)
Citation Context ... Even for second-order elliptic operators, the investigation of the imaginary-time heat kernel (that is, the fundamental solution of the Schro� dinger equation) is a very hard problem. It is shown in =-=[46, 104]-=- that if the second-order coefficients converge to those of �∆ sufficiently rapidly at infinity, and no point in the phase space T*(RN) is trapped forwards or backwards by the bicharacteristic flow, t... |

49 |
dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt
- Faber, Beweis
- 1923
(Show Context)
Citation Context ...ely upon the shape of the horn and on the boundary conditions [100, 119, 120, 27, 28, 63]. 2.2 The isoperimetric problem. In 1877 Rayleigh made two conjectures, the first of which was solved by Faber =-=[79]-=- in 1923; this was the matter of proving that the plane region of unit area for which the Laplacian subject to Dirichlet boundary conditions has minimum principal eigenvalue is a circular disc, and is... |

49 |
Analysis and geometry on groups, Cambridge Tracts in Mathematics 100 Cambridge
- Varopoulos, Saloff-Coste, et al.
- 1992
(Show Context)
Citation Context .... IfHis second-order elliptic acting in L�(RN), then there are Gaussian lower bounds on the heat kernel which complement the upper bounds, provided that the coefficients of the operator are real; see =-=[168, 50, 152]-=- for the extensive literature on this problem, and its generalization to Riemannian manifolds M under suitable geometrical conditions on the latter. If M is noncompact, then a principal object of inte... |

44 |
On the closedness of the sum of two closed operators
- Dore, Venni
(Show Context)
Citation Context ...complex powers Az and Bz are sufficiently wellbehaved. It turns out that the theory depends upon � being a ζ-convex space, in the sense that the Hilbert transform is bounded on Lp(R,�) for 1�p��. See =-=[6, 65]-=- for more details concerning the proper interpretation of the above formula and its applications. We conclude by mentioning that other aspects of the theory of the H � functional calculus also depend ... |

40 |
Existence theorems for trapped modes
- Evans, Levitin, et al.
- 1994
(Show Context)
Citation Context ... other papers cited there for examples in which essential spectrum and even absolutely continuous spectrum can occur specifically because of the choice of Neumann boundary conditions. We also mention =-=[76, 62]-=- for new results concerning the spectral theory of acoustic waveguides, for which the physically appropriate boundary conditions are the Neumann ones. Our goal is to review results obtained within the... |

38 | Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
- Grunau, Sweers
(Show Context)
Citation Context ...r G(x,y), which is also visibly positive, may be found in [91]. The formula and explicit estimates of the rate of vanishing of G(x,y) and its partial derivatives as x��Bare given by Grunau and Sweers =-=[90]-=-; they also obtain similar results for the polyharmonic operator subject to a sufficiently small lower-order perturbation. Typical of their results is the uniform estimate in terms of d(x)�1��x� for N... |

37 |
Shaposhnikova; Theory of Multipliers in Spaces of Differentiable Functions
- Maz’ya, O
- 1985
(Show Context)
Citation Context ...undary of Ω is regular in the sense that it is determined by a differentiable function such that the modulus of continuity ω(t) of its gradient satisfies ω(t)�t�Lp(0,1), then g�Lp(Ω) implies f�W�,p(Ω)=-=[131]-=-; there is also an improvement and extension of this result to higher-order operators. However, if 2m�4 and �Ω is piecewise smooth, then f may be unbounded at a corner of �Ω even if g�C � (Ω� ). Our p... |

36 | The functional calculus - Davies |

35 |
M.Z.Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations
- Birman
- 1991
(Show Context)
Citation Context ...erators for which the potential is negative and converges to zero at infinity. Much of this concentrates on inequalities satisfied by N(λ) for all λ�0, and the results are necessarily simpler if N�2m =-=[31, 33]-=-. We finally refer to [140] for Lieb–Thirring type inequalities satisfied by the negative eigenvalues of such generalized Schro� dinger operators. Finally, there are cases of degenerate or irregular e... |

34 |
Explicit constants for Gaussian upper bounds on heat kernels
- Davies
- 1987
(Show Context)
Citation Context ...acts in L�(Ω), where Ω is non-convex, then it is better to take d to be the infimum of the length of curves from x to y which stay entirely inside Ω. The possibility of doing this was demonstrated in =-=[49]-=- for the Laplacian, with the obvious generalization for other second-order elliptic operators [50, Theorem 3.2.7]. Of course, the choice of c is also important, and the same papers showed that one � c... |

34 |
Non-Gaussian aspects of heat kernel behaviour
- Davies
- 1997
(Show Context)
Citation Context ...eral properties of heat kernels. Most of the results below are specific to uniformly elliptic operators acting in L�(RN), but the following conclusions can be derived by an extremely general analysis =-=[56]-=-. Suppose that K is the heat kernel of an abstract non-negative self-adjoint operator H acting in L�(M,dx), where dx is a regular Borel measure on a locally compact metric space M. The function t�K(t,... |

32 |
Banach space operators with a bounded H∞ functional calculus
- Cowling, Doust, et al.
- 1996
(Show Context)
Citation Context ...s in Banach spaces, to which we now turn. 7.3 H � functional calculus. We give a short and slightly over-simplified account of McIntosh’s theory of the H � functional calculus, following particularly =-=[133, 5, 7, 43]-=-. Let H be an abstract unbounded operator with dense domain in a Banach space � and with spectrum contained in the sector for some 0�β�π. Ifγ�β, put S β��z: �arg(z)��β���0� (7.3.1) S� γ ��z�0:�arg(z)�... |

31 |
Semigroup kernels, Poisson bounds and holomorphic functional calculus
- Duong, Robinson
- 1996
(Show Context)
Citation Context ...-order coefficients of H are uniformly continuous, then there is a large literature on elliptic regularity problems [3, 4, 138, 83], Lp properties of the semigroup e−Ht and the Lp functional calculus =-=[7, 5, 67, 68]-=-. Some aspects of the self-adjoint theory need only small modifications, but there are new problems which do not arise in the self-adjoint case. In the next subsection we discuss the second-order case... |

30 |
Elliptic Operators and Lie Groups
- Robinson
- 1991
(Show Context)
Citation Context ...s to Lp. IfN�2m, then the semigroup may be extended to Lp for all 1�p�� even if the coefficients are measurable. If the coefficients are in C �, then b the same holds without restriction on the order =-=[108, 152]-=-. It is interesting that for 2m�4, the proofs of these facts depend upon first obtaining appropriate kernel bounds, for example those discussed in the next section, whereas for 2m�2 the Lp property is... |

29 |
On a question of Hadamard concerning super-biharmonic functions
- DUFFIN
- 1949
(Show Context)
Citation Context ...etre [75]. Their formula enables them to show that the Green function is non-positive for all 0�ε �1. The corresponding conjecture for convex regions was proved false for the infinite strip by Duffin =-=[66]-=- long before it was known that the ground state of the unit square is of variable sign. An interesting historical discussion of this problem was given by Hedenmalm [92], who also obtained explicit Gre... |

26 | Trapped modes in acoustic waveguides
- Davies, Parnovski
- 1998
(Show Context)
Citation Context ... other papers cited there for examples in which essential spectrum and even absolutely continuous spectrum can occur specifically because of the choice of Neumann boundary conditions. We also mention =-=[76, 62]-=- for new results concerning the spectral theory of acoustic waveguides, for which the physically appropriate boundary conditions are the Neumann ones. Our goal is to review results obtained within the... |

26 | The essential spectrum of Neumann Laplacians on some bounded singular domains
- Hempel, Seco, et al.
- 1991
(Show Context)
Citation Context ...eometrical’ conditions on the region for the Neumann Laplacian to have compact resolvent; here ‘geometrical’ means conditions involving bounds on capacities, etc., of all compact subsets. We refer to =-=[63, 94, 100]-=- and other papers cited there for examples in which essential spectrum and even absolutely continuous spectrum can occur specifically because of the choice of Neumann boundary conditions. We also ment... |

25 |
Regularity theorems and heat kernel for elliptic operators
- Auscher
- 1996
(Show Context)
Citation Context ...14, 15, 74]. However, if one assumessLp spectral theory 533 Ho� lder continuous or more regular coefficients, then the dimensional restriction becomes entirely unnecessary [108, 152, 13, 14]. Auscher =-=[12]-=- has extended this result to the case of uniformly continuous coefficients by the use of Morrey spaces, but bounded continuous coefficients do not suffice. No such theorem can be proved for N�2m�4 if ... |

24 |
Gaussian estimates and interpolation of the spectrum
- Arendt
- 1994
(Show Context)
Citation Context ...perbolic space Hn+� is the complex region on or inside the parabola t�R�n�p−�(1�p−�)�t��itn(1�2�p). (6.1.2) Other simple examples of operators for which the Lp spectrum depends upon p may be found in =-=[8]-=-. The best result in the reverse direction for Laplace–Beltrami operators is that of Sturm, who proved that if large balls of the manifold have uniformly subexponential volume growth, then one has p-i... |

24 |
Uniformly elliptic operators with measurable coefficients
- Davies
- 1995
(Show Context)
Citation Context ...rt III] � states that this is contained in a space C α � (Ω� )ofHo�lder continuous functions on Ω� which vanish on the boundary �Ω. Therefore g�L� implies f�C α (Ω� ). It also follows from results in =-=[52]-=- that H−� is a bounded operator on Lp for all p�[1,�], and that it has a continuous integral kernel which vanishes on the boundary of Ω. Therefore g�Lp for any 1�p�� implies f�Lp, the solution f being... |

24 |
Über Minimaleigenschaften der Kugel in drei un mehr Dimensionen
- Krahn
- 1926
(Show Context)
Citation Context ...ect to Dirichlet boundary conditions has minimum principal eigenvalue is a circular disc, and is known as the isoperimetric problem for a membrane. The higher-dimensional analogue was solved by Krahn =-=[115, 116]-=- in 1925�26. The second conjecture was that among all clamped plates with a given area, the circular disc again has the minimal principal eigenvalue λ � (Ω). The clamped plate may be identified mathem... |

22 |
Operators which have an H∞ functional calculus. In Miniconference on operator theory and partial differential equations (North Ryde
- McIntosh
- 1986
(Show Context)
Citation Context ...s in Banach spaces, to which we now turn. 7.3 H � functional calculus. We give a short and slightly over-simplified account of McIntosh’s theory of the H � functional calculus, following particularly =-=[133, 5, 7, 43]-=-. Let H be an abstract unbounded operator with dense domain in a Banach space � and with spectrum contained in the sector for some 0�β�π. Ifγ�β, put S β��z: �arg(z)��β���0� (7.3.1) S� γ ��z�0:�arg(z)�... |

20 | Explicit constants for Rellich inequalities
- Davies, Hinz
- 1998
(Show Context)
Citation Context ...stant c and all f�Wm,� [127, Theorems 2.5.2, 12.5.1; 132]. Sharp constants for such Rellich type inequalities in Lp were obtained in [109] for Euclidean space using rearrangement inequalities, and in =-=[59]-=- by a method applicable to Riemannian manifolds. In particular, the sharp constant c for � RN �u(x)�p �x�β +�mp dx�c� �∆mu(x)�p RN �x�β dx (2.3.8) is known for all N,β,m,p. Such inequalities play an i... |

18 |
On-diagonal lower bounds for heat kernels and Markov chains
- Coulhon, Grigor’yan
- 1997
(Show Context)
Citation Context ...g property vol(x,2r)�cvol(x,r) (5.5.2) for some c�0 and all x�M, r�0; see [87, 152, 156, 168] for references to some of the enormous recent literature on this problem. We draw particular attention to =-=[40]-=-, where diagonal lower bounds on K(t,x,x) are obtained without recourse to the use of the parabolic Harnack inequality, which is an intrinsically second-order property. It is a classical result that i... |

18 |
On sign variation and the absence of "strong" zeros of solutions of elliptic equations
- KozLov, KONDRAT’EV, et al.
- 1990
(Show Context)
Citation Context ...740�1.119i, one would expect � f to be of the form � f (x,x)�Re�cxp+�� (2.5.4) � near the origin unless the first coefficient in its expansion vanishes.s522 e. b. davies Kozlov, Kondrat’ev and Maz’ya =-=[112]-=- obtained similar results for eigenfunctions of a class of constant-coefficient higher-order elliptic operators in higher space dimensions at conical vertices with sufficiently small angles. They also... |

17 |
On the Lp-spectrum of uniformly elliptic operators on Riemannian manifolds
- Sturm
(Show Context)
Citation Context ...erse direction for Laplace–Beltrami operators is that of Sturm, who proved that if large balls of the manifold have uniformly subexponential volume growth, then one has p-independence of the spectrum =-=[165]-=-. The method which we describe here can be extended to complete Riemannian manifolds for which the volumes of large balls are polynomially bounded, but we confine the discussion to the case of Euclide... |

16 |
Heat kernel bounds on hyperbolic space and Kleinian groups
- Davies, Mandouvalos
- 1988
(Show Context)
Citation Context ... − the Lp spectrum of an elliptic operator H is independent of p, even if we restrict attention to the Laplace–Beltrami operator of a complete Riemannian manifold of bounded geometry. It was shown in =-=[124, 61]-=- that for p�2, the Lp spectrum of hyperbolic space Hn+� is the complex region on or inside the parabola t�R�n�p−�(1�p−�)�t��itn(1�2�p). (6.1.2) Other simple examples of operators for which the Lp spec... |

16 |
Dilation invariant estimates and the boundary G̊arding inequality for higher order elliptic operators
- Pipher, Verchota
(Show Context)
Citation Context ...ion in bounded domains in RN with Lipschitz boundaries when N�2,3, but not for N�4. The most complete current results, for general higherorder constant-coefficient elliptic operators, may be found in =-=[150]-=-. Counterexamples to some of the conjectures are obtained by considering regions with conical points, as described in the next subsection; see also [113]. 3.6 Conical points. Many of the crucial examp... |