Fredholm property of general elliptic problems

Volpert, A. I.; Volpert, Vitaly

doi:10.1090/s0077-1554-06-00159-2

Cited by 15 publications

(10 citation statements)

References 33 publications

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“…Muhamadiev's work has been a source of inspiration for the decades that followed. For example, similar to his main results in [73] but much more recently, A. and V. Volpert show that, for a rather general class of scalar elliptic partial differential operators L on rather general unbounded domains and also for systems of such, a Favard condition is equivalent to the Φ + property of L on appropriate Hölder [109,110,111] or Sobolev [108,110,111] spaces.…”

Section: A Brief Historysupporting

confidence: 54%

Limit operators, collective compactness, and the spectral theory of infinite matrices

2011

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In the first half of this text we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann [85] and Lindner [63]) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang [26]. We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator A (its operator spectrum). In the second half of this text we study bounded linear operators on the generalised sequence space p (Z N , U ), where p ∈ [1, ∞] and U is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator A is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of p = 1 and ∞. Our tools in this study are the results from the first half of the text and an exploitation of the partial duality between 1 and ∞ and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the text. Results in this second half of the text include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both selfadjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on R N .

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Section: A Brief Historysupporting

confidence: 54%

Limit operators, collective compactness, and the spectral theory of infinite matrices

2011

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“…We deduce easily that either u ∈ U or u(x 1 , Proof. There exist results for the Fredholm property for general linear elliptic problems in unbounded domains [20]. Let us give the proof that we did for this particular problem.…”

Section: Let Us Prove That the Convergence Is Uniform Inmentioning

confidence: 93%

Singly periodic solutions of a semilinear equation

Allain¹,

Beaulieu²

2009

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider the solutions of the equation −ε 2 u + u − |u| p−1 u = 0 in S 1 × R, where ε and p are positive real numbers, p > 1. We prove that the set of the positive bounded solutions even in x 1 and x 2 , decreasing for x 1 ∈ ]−π, 0[ and tending to 0 as x 2 tends to +∞ is the first branch of solutions constructed by bifurcation from the ground-state solution (ε, w 0 ( x 2 ε )). We prove that there exists a positive real number ε such that for every ε ∈]0, ε ] there exists a finite number of solutions verifying the above properties and none such solution for ε > ε . The proves make use of compactness results and of the Leray-Schauder degree theory. RésuméNous étudions l'équation −ε 2 u + u − |u| p−1 u = 0 dans S 1 × R, où ε et p sont des nombres réels strictement positifs, p > 1. Nous identifions l'ensemble des solutions (ε, u) où u est une fonction positive, paire en x 1 et x 2 , décroissante en x 1 dans [−π, 0] et tendant vers 0 quand x 2 tend vers +∞, comme la première branche de solutions issue d'une bifurcation à partir de l'état fondamental (ε, w 0 ( x 2 ε )). Nous prouvons qu'il existe un réel ε tel que pour tout ε ∈ ]0, ε ] il y a un nombre fini de solutions vérifiant les propriétés énoncées ci-dessus, et aucune telle solution pour ε > ε . Les preuves utilisent des résultats de compacité et la théorie du degré de Leray-Schauder.

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“…31, 32 In this case, mass-action kinetics reactions are again represented by species-reaction networks with edge colouring (or equivalently the directed edges are weighted by the stoichiometric coefficients). A feedback cycle in a network is a subnetwork where each species node is the beginning and end of a path.…”

Section: Network Analysis Methods For Investigating the Dynamic Behavmentioning

confidence: 99%

Network representations and methods for the analysis of chemical and biochemical pathways

2013

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Systems biologists increasingly use network representations to investigate biochemical pathways and their dynamic behaviours. In this critical review, we discuss four commonly used network representations of chemical and biochemical pathways. We illustrate how some of these representations reduce network complexity but result in the ambiguous representation of biochemical pathways. We also examine the current theoretical approaches available to investigate the dynamic behaviour of chemical and biochemical networks. Finally, we describe how the critical chemical and biochemical pathways responsible for emergent dynamic behaviour can be identified using network mining and functional mapping approaches.

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Fredholm property of general elliptic problems

Cited by 15 publications

References 33 publications

Limit operators, collective compactness, and the spectral theory of infinite matrices

Limit operators, collective compactness, and the spectral theory of infinite matrices

Singly periodic solutions of a semilinear equation

Network representations and methods for the analysis of chemical and biochemical pathways

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