Construction of the Leray-Schauder degree for elliptic operators in unbounded domains (original) (raw)

Pointwise estimates for a class of strongly degenerate elliptic operators : a geometrical approach

1987

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Bounded R00-CALCULUS for Elliptic Operators

2013

It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. 0. Introduction. It is the main purpose of this paper to prove under mild regularity assumptionsthat Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers. In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8&0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0 , n), we ...

Leray-Schauder degree: a half century of extensions and applications

Topological Methods in Nonlinear Analysis, 1999

The Leray-Schauder degree is defined for mappings of the form I − C, where C is a compact mapping from the closure of an open bounded subset of a Banach space X into X. Since the fifties, a lot of work has been devoted in extending this theory to the same type of mappings on some nonlinear spaces, and in extending the class of mappings in the frame of Banach spaces or manifolds. New applications of Leray-Schauder theory and its extensions have also been given, specially in bifurcation theory, nonlinear boundary value problems and equations in ordered spaces. The paper surveys those developments.

Spectral theory of elliptic operators in exterior domains

Russian Journal of Mathematical Physics, 2010

We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type A = P 0 |α|,|β| m (−1) α D α a α,β (x)D β , a α,β (·) ∈ C ∞ (Ω), on smooth (bounded or unbounded) domains Ω in R n with compact boundary ∂Ω. Using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of these closed realizations of A and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization.

The Leray-Schauder Degree as Topological Method Solution of Nonlinear Elliptic Equations

Pure and Applied Mathematics Journal, 2017

In the present paper using precise results on the solutions of linear elliptic differential operators with Holder continuous coefficient as well as a variant of the Lery-Schauder method and the gal of this paper to find an adequate degree theory for the infinite dimensional setting and to extend the theory of homotopy classes of maps form ℝ to ℝ to homotopy classes of maps on infinite dimensional spaces.

Degenerate elliptic operators: capacity, flux and separation, 2005

Let S = {S t } t≥0 be the semigroup generated on L 2 (R d ) by a selfadjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R d with Lipschitz continuous boundary ∂Ω. We prove that S leaves L 2 (Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.

A unified approach to universal inequalities for eigenvalues of elliptic operators

Pacific Journal of Mathematics, 2004

We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

The Maslov index and the spectra of second order elliptic operators

Advances in Mathematics

We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H 1 (Ω), and Lagrangian planes in H 1/2 (∂Ω) × H −1/2 (∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H 1/2 (∂Ω)× H −1/2 (∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ−periodic Schrödinger operators on a period cell Q ⊂ R n , the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on starshaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H 1 (Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators. Contents 1. Introduction 2 2. Self-adjoint extensions and Lagrangian planes 6 2.1. Assumptions 6 2.2. The Lagrangian planes and the self-adjoint extensions of differential operators 9 2.3. The Maslov index 13 3. The Maslov index for second order elliptic operators on smooth domains 17 3.1. Weak solutions and their traces 17 3.2. The Maslov and Morse indices 21 3.3. The spectral flow and the Maslov index 26 3.4. Spectra of elliptic operators on deformed domains and the Maslov index 26 3.5. Spectra of elliptic operators with Robin boundary conditions and the Maslov index 28 4. The Maslov index for the Schrödinger operators on Lipschitz domains 30 4.1. A general result for the Schrödinger operators 30 4.2. Spectra of θ−periodic Schrödinger operators and the Maslov index 33 4.3. Spectra of Schrödinger operators on star-shaped domains 36

A spectral theory for order unit spaces

Annales De L Institut Henri Poincare-physique Theorique, 1981

L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 259