Properness and topological degree for general elliptic operators

Volpert, Vitaly; Volpert, A. I.

doi:10.1155/s1085337503204073

Cited by 32 publications

(48 citation statements)

References 25 publications

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“…Topological degree for elliptic operators in unbounded domains is constructed using the properties of Fredholm and proper operators with the zero index [13,14,15]. The same construction can be used for the nonlocal reaction-diffusion operators.…”

Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

“…The method to obtain a priori estimates of solutions is similar to the method developed for monotone reaction-diffusion systems [15]. It is based on the maximum principle which is applicable for the equations under consideration.…”

Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

“…By definition, topological degree is an integer number γ(A, D) which depends on the operator and on the domain D in the function space E. Topological degree for elliptic operators in unbounded domains is constructed in [15,14] on the basis of the theory of Fredholm and proper operators. The results on the Fredholm property and properness of the integro-differential operators presented above allow us to use the same construction.…”

Section: Construction Of Topological Degreementioning

confidence: 99%

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Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Demin

Volpert

2010

Math. Model. Nat. Phenom.

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Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

Section: Construction Of Topological Degreementioning

confidence: 99%

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Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Demin

Volpert

2010

Math. Model. Nat. Phenom.

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“…In the general case limiting operators and domains are introduced in [44], [45]. Their construction can be briefly described as follows.…”

Section: Limiting Problemsmentioning

confidence: 99%

“…This is a necessary and sufficient condition for general elliptic operators considered in Hölder spaces to be normally solvable with a finite-dimensional kernel [44]. For scalar elliptic problems in Sobolev spaces it was proved in [45].…”

Section: Condition Ns Any Limiting Problemmentioning

confidence: 99%

Fredholm property of general elliptic problems

Volpert

2006

Trans. Moscow Math. Soc.

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Abstract. Linear elliptic problems in bounded domains are normally solvable with a finite-dimensional kernel and a finite codimension of the image, that is, satisfy the Fredholm property, if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii condition are satisfied. In the case of unbounded domains these conditions are not sufficient any more. The necessary and sufficient conditions of normal solvability with a finite-dimensional kernel are formulated in terms of limiting problems. Adjoint operators to elliptic operators in unbounded domains are studied and the conditions in order for them to be normally solvable with a finite-dimensional kernel are also formulated by means of limiting problems. The properties of the direct and of the adjoint operators are used to prove the Fredholm property of elliptic problems in unbounded domains. Some special function spaces introduced in this work play an important role in the study of elliptic problems in unbounded domains.

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On properness and related properties of quasilinear systems on unbounded domains

Krömer

Lilli

2011

Mathematische Nachrichten

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Key words Proper nonlinear operators, quasilinear systems, compactness, decomposition lemma MSC (2010) 35G30, 35J55, 35A35The purpose of this paper is to provide tools for analyzing the compactness properties of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general class of nonlinear operators arising in quasilinear systems of partial differential equations of second order, in divergence form. Our approach, based on a suitable decomposition lemma, admits the discussion of problems with some inherent loss of compactness, for example due to a domain with infinite measure or a lower order term with critical growth. As an application, we obtain a characterization of properness which is considerably easier to verify than the definition.

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Properness and topological degree for general elliptic operators

Abstract: The paper is devoted to general elliptic operators in Hölder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and proper operators of index zero.

Cited by 32 publications

References 25 publications

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Fredholm property of general elliptic problems

On properness and related properties of quasilinear systems on unbounded domains

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