EUDML

The lecture presents current results on heat trace expansions, and the related resolvent trace and zeta function expansions, for elliptic operators with boundary conditions on $n$ -dimensional compact manifolds. As a background, we recall the set-up of elliptic differential operators with differential boundary conditions having heat trace expansions in powers $\sum_{k \geq - n} c_{k} t^{k / m}$ . Then we consider the spectral boundary conditions of Atiyah, Patodi and Singer for Dirac-type first-order operators, leading to expansions...

Weakly semibounded boundary problems and sesquilinear forms

Gerd Grubb — 1973

Annales de l'institut Fourier

Let $A$ be a $2 m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\overline{Ω}$ with boundary $Γ$ ; and denote by $A_{B}$ the realization defined by a normal differential boundary condition $B ρ u = 0$ ( $u \in H^{2 m} (E)$ , $ρ u =$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $Γ$ , the realizations $A_{B}$ for which there exists an integro-differential sesquilinear form $a_{B} (u, ν)$ on $H^{m} (E)$ such that $(A u, ν) = a_{B} (u, ν)$ on $D (A_{B})$ ; moreover we show that these are exactly the realizations satisfying a weak semiboundedness estimate:...

The Essential Spectrum of Elliptic Systems of Mixed Order.

Gerd Grubb; Giuseppe Geymonat — 1977

Mathematische Annalen

The transmission property.

Lars Hörmander; Gerd Grubb — 1990

Mathematica Scandinavica

Traces and quasi-traces on the Boutet de Monvel algebra

Gerd Grubb; Elmar Schrohe — 2004

Annales de l’institut Fourier

We construct an analogue of Kontsevich and Vishik’s canonical trace for pseudodifferential boundary value problems in the Boutet de Monvel 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)...

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Parameter-elliptic and parabolic pseudodifferential boundary problems in global Lp Sobolev spaces.

Remainder estimates for eigenvalues and kernels of pseudo-differential elliptic systems.

Sur la théorie spectrale des problèmes aux limites pseudo-différentiels elliptiques

Sur la résolvante et le calcul fonctionnel des problèmes aux limites pseudo-différentiels

Le problème d'Atiyah-Patodi-Singer pour les variétés à bord générales (non cylindriques)

Singular perturbations and parameter-dependent pseudo-differential boundary problems

On the spectral theory of pseudo-differential elliptic boundary problems

A characterization of the non local boundary value problems associated with an elliptic operator

Properties of normal boundary problems for elliptic even-order systems

Conditions au bord spectrales et formules de trace

Weakly semibounded boundary problems and sesquilinear forms

The Essential Spectrum of Elliptic Systems of Mixed Order.

The transmission property.

Traces and quasi-traces on the Boutet de Monvel algebra

Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods.

Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems.

A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equation