arXiv subjects

Gerd Grubb

Publications and source records attributed to Gerd Grubb.

At least 37 records · Page 2Linked to original sources

arXiv2014

Spectral results for mixed problems and fractional elliptic operators

In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{χ,Σ_+}$ in $L_2(Ω)$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $Ω\subset R^n$; here the boundary $\partialΩ=Σ$ is partioned smoothly into $Σ=Σ_-\cup Σ_+$, the Dirichlet condition $γ_0u=0$ is imposed on $Σ_-$, and a Neumann or Robin condition $χu=0$ is imposed on $Σ_+$. It is shown that the Dirichlet-to-Neumann operator $P_{γ,χ}$ is principally of type $\frac12$ with factorization index $\frac12$, relative to $Σ_+$. The above theory allows a detailed description of $D(A_{χ,Σ_+})$ with singular elements outside of $H^{\frac32}(Ω)$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{χ,Σ_+}^{-1}-A_γ^{-1}$.

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arXiv2014

Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with $B^\frac32_{p,2}$-smoothness and operators with $H^1_q$-coefficients, for suitable $p>2(n-1)$ and $q>n$. In particular, Kre\uın-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.

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arXiv2013

Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.

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arXiv2013

Spectral asymptotics for nonsmooth singular Green operators

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω\subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**) s_j(G) \le Cj^{-t/(n-1)} are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t=2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in W^1_p(Ω) for some p>n.

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arXiv2011

Krein-like extensions and the lower boundedness problem for elliptic operators

For selfadjoint extensions tilde-A of a symmetric densely defined positive operator A_min, the lower boundedness problem is the question of whether tilde-A is lower bounded {\it if and only if} an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets 1976); this applies to elliptic operators A on bounded domains. For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for general T. The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann extension A_0, appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .

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arXiv2011

Extension theory for elliptic partial differential operators with pseudodifferential methods

This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very useful here, not only as a formulational framework, but also for the solution of specific questions. We recall some elements of that theory, and show its application in several cases (including recent results), namely to the lower boundedness question, and the question of spectral asymptotics for differences between resolvents.

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arXiv2011

The sectorial projection defined from logarithms

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,ξ) have arguments in \,]θ,ϕ[\, and \,]ϕ, θ+2π[\, (θ<ϕ<θ+2π), the sectorial projection Π_{θ, ϕ}(P) is defined essentially as the integral of the resolvent along {e^{iϕ}R_+}\cup {e^{iθ}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that ¶_{θ, ϕ}(P) is a ψdo of order 0; namely that p_m(x,ξ) cannot in general be modified to allow integration of (p_m(x,ξ)-λ)^{-1} along {e^{iϕ}R_+}\cup {e^{iθ}R_+} simultaneously for all ξ. We show that the structure of Π_{θ, ϕ}(P) as a ψdo of order 0 can be deduced from the formula Π_{θ, ϕ}(P)= (i/(2π))(\log_θ(P) - \log_ϕ(P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_θ(P) one need only modify p_m(x,ξ) in a neighborhood of e^{iθ}R_+; this is known to be possible from Seeley's 1967 work on complex powers.

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arXiv2011

The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set Ωin R^n with boundary Σ, the mixed problem is defined by a Neumann-type condition on a part Sigma_+ of the boundary and a Dirichlet condition on the other part Sigma_-. We show a Krein resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Sigma_+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s_j j^{2/(n-1)}\to C_{0,+}^{2/(n-1)}, where C_{0,+} is proportional to the area of Sigma_+, in the case where A is principally equal to the Laplacian.

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arXiv2011

Spectral asymptotics for Robin problems with a discontinuous coefficient

The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $νu=b_1γ_0u$ and $νu=b_2γ_0u$, can in the case where all coefficients are $C^\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -λ)^{-1}-(\tilde A_2 -λ)^{-1}$ satisfy $s_j j^{3/(n-1)}\le C$ for all $j$; this improves a recent result for $A=-Δ$ by Behrndt et al., that $\sum_js_j ^p<\infty$ for $p>(n-1)/3$. A sharper estimate is obtained when $b_1$ and $b_2$ are in $C^ε$ for some $ε>0$, with jumps at a smooth hypersurface, namely that $s_j j^{3/(n-1)}\to c$ for $j\to \infty$, with a constant $c$ defined from the principal symbol of $A$ and $b_2-b_1$. As an auxiliary result we show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators interspersed with piecewise continuous functions.

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arXiv2009

Perturbation of essential spectra of exterior elliptic problems

For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Diriclet data to Neumann data. We here also show an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. - The proofs rely on Krein-type formulas for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.

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arXiv2008

Krein resolvent formulas for elliptic boundary problems in nonsmooth domains

The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols.

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arXiv2007

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) -- for example the solution operator of a classical elliptic problem -- and $P_{1,T}$ is a realization of an elliptic differential operator $P_1$, having a ray free of eigenvalues. Relative formulas (e.g. for the difference between the constants with two different choices of $P_{1,T}$) have been known for some time and are local. We here determine $C_0(B, P_{1,T})$ itself, showing how it is put together of local residue-type integrals (generalizing the noncommutative residue of Wodzicki, Guillemin, Fedosov-Golse-Leichtnam-Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes that of Paycha and Scott, shown recently for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving $\log P_{1,T}$. Since the complex powers of $P_{1,T}$ lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.

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arXiv2007

Logarithms and sectorial projections for elliptic boundary problems

On a compact manifold with boundary, consider the realization B of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say R_-. In the first part of the paper we define and discuss in detail the operator log B; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, zeta(B,0), and it enters in an important way in studies of composed zeta functions zeta(A,B,s)=Tr(AB^{-s}) (pursued elsewhere). There is a similar definition of the operator log_theta B, when the spectral cut is at a general angle theta. When B has spectral cuts at two angles theta < phi, one can define the sectorial projection Pi_{theta,phi}(B) whose range contains the generalized eigenspaces for eigenvalues with argument in ] theta, phi [; this is studied in the last part of the paper. The operator Pi_{theta,phi}(B) is shown to be proportional to the difference between log_theta B and log_phi B, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.

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arXiv2005

A resolvent approach to traces and zeta Laurent expansions

Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C_0(A,P) in the trace expansion of A(P-λ)^{-N} (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A is extended from the cases of noninteger order, or integer order and even-even parity on odd-dimensional manifolds, to the case of even-odd parity on even-dimensional manifolds. For the generalized zeta function ζ(A,P,s)=\Tr(AP^{-s}), extended meromorphically to C, C_0(A,P) equals the coefficient of s^0 in the Laurent expansion at s=0 when P is invertible. In the mentioned parity cases, ζ(A,P,s) is regular at all integer points. The higher Laurent coefficients C_j(A,P) at s=0 are described as leading nonlocal coeficients C_0(B,P) in trace expansions of resolvent expressions B(P-λ)^{-N}, with B log-polyhomogeneous as defined by Lesch (here -C_1(I,P)=C_0(\log P,P) gives the zeta-determinant). C_0(B,P) is shown to be a quasi-trace in general, a canonical trace TR B in restricted cases, and the formula of Lesch for TR B in terms of a finite part integral of the symbol is extended to the parity cases.

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arXiv2005

Remarks on nonlocal trace expansion coefficients

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a formula for the zero'th coefficient at s=0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). - We also give some corrections to transition formulas used in our earlier works.

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arXiv2004

On the logarithm component in trace defect formulas

In asymptotic expansions of resolvent traces $\Tr(A(P-λ)^{-1})$ for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-λ)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.

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arXiv2004

Traces and Quasi-traces on the Boutet de Monvel Algebra

We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-λ)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-λ)^{-N}) (with N large) in powers and log-powers of λas λtends to infinity in a suitable sector of the complex plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series for the meromorphic extension of the generalized zeta function ζ(A,B,s)= Tr(AB^{-s}) at s=0, when B is invertible. We show that C_0(A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A'] modulo local terms, and has a specific value independent of B modulo local terms; and we single out particular cases where the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A. Our main tool is a precise analysis of the asymptotic expansion of the resolvent trace, based on pseudodifferential calculations involving rational functions (in particular Laguerre functions) of the normal variable.

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arXiv2004

Analysis of Invariants Associated with Spectral Boundary Problems for an Elliptic Operator

This is a survey of recent results on zeta- and eta-function poles and values for realizations of Laplace- and Dirac-type operators defined by pseudodifferential projection boundary conditions (including the Atiyah-Patodi-Singer operator and its square). Section 1 recalls some useful results for ps.d.o.s on closed manifolds. Section 2 describes the asymptotic trace expansions for second-order operators under general projection boundary conditions, in particular the vanishing of the zeta residue at 0 in general (with consequences for first-order operators). Section 3 treats cases with symmetry or selfadjointness conditions, where a stability of the zeta value at 0, or of the vanishing of the eta residue, can be shown for low-order perturbations of the boundary projection. Section 4 describes general results on stability of expansion coefficients when the interior operator is perturbed, and ends with special results obtainable when the perturbation commutes with the interior operator near the boundary, in a suitable sense.

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