Elliptic Equations in Unbounded Domains

Bagirov, L. A.

doi:10.1070/sm1971v015n01abeh001535

Cited by 7 publications

(5 citation statements)

References 4 publications

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“…Limiting operators and their interrelation with solvability conditions and with the Fredholm property were first studied in [15], [20], [21] (see also [39]) for differential operators on the real axis, and later for some classes of elliptic operators in R n [8], [25], [26], in cylindrical domains [9], [43], or in some specially constructed domains [6], [7]. Some of these results are obtained for the scalar case, some others for the vector case, under the assumption that the coefficients of the operator stabilize at infinity or without this assumption.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

Fredholm property of general elliptic problems

Volpert

2006

Trans. Moscow Math. Soc.

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“…The proofs of Theorem 3.1 and Theorem 3.3 from [20] for p = 2 can be generalized for the spaces H k+1,R,p q (R n ) in a similar way. □ Further we establish that the regularity of P(x, D) in R n and condition (10) are not only the necessary conditions but also sufficient for the fulfillment of the a priori estimate (9) in spaces H k,R,p q (R n ). Lemma 3.1 Let k ∈ Z + , q ∈ Q k,R , x 0 ∈ R n and P 0 (x, D) be the differential form (5).…”

Section: A Priori Estimates In Multianisotropic Spacesmentioning

confidence: 86%

“…Using the last inequality and Lemma 3.1, similarly to the proof of Theorem 2.2 from the work [13], it is easy to check that for a big enough m 0 operators P m (x, D) : H k+1,R,p q (R n ) → H k,R,p q (R n ) for m > m 0 have bounded inverse operators. Since (10) holds they have uniformly bounded norms and with some C 2 > 0 the following holds…”

Section: Hencementioning

confidence: 99%

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Fredholm property of regular hypoelliptic operators on the scales of multianisotropic spaces

Tumanyan

2022

ITM Web Conf.

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This paper studies the Fredholm properties for a class of regular hypoelliptic operators. We establish necessary and suﬃcient conditions for a priori estimates for diﬀerential operators acting in multianisotropic Sobolev spaces in Rn. Fredholm criteria and index invariance are obtained for a wide class of regular hypoelliptic operators on the special scales of multianisotropic weighted spaces.

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“…Since we do not need general dimension in this paper, we omit them here. For interested readers, one may refer to [30] for the pointwise estimates at infinity and to, for example, [31,32,30] for the theories of elliptic equations in unbounded domains.…”

Section: Basic Setup Of Sdesmentioning

confidence: 99%

Large time behaviors of upwind schemes by jump processes

Li¹,

Liu²

2018

Preprint

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We revisit the traditional upwind schemes for linear conservation laws in the viewpoint of jump processes, allowing studying upwind schemes using probabilistic tools. In particular, for Fokker-Planck equations on R, in the case of weak confinement, we show that the solution of upwind scheme converges to a stationary solution. In the case of strong confinement, using a discrete Poincaré inequality, we prove that the O(h) numeric error under 1 norm is uniform in time, and establish the uniform exponential convergence to the steady states. Compared with the traditional results of exponential convergence of upwind schemes, our result is in the whole space without boundary. We also establish similar results on torus for which the stationary solution of the scheme does not have detailed balance. This work shows an interesting connection between standard numerical methods and time continuous Markov chains, and could motivate better understanding of numerical analysis for conservation laws.

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Elliptic Equations in Unbounded Domains

Cited by 7 publications

References 4 publications

Fredholm property of general elliptic problems

Fredholm property of general elliptic problems

Fredholm property of regular hypoelliptic operators on the scales of multianisotropic spaces

Large time behaviors of upwind schemes by jump processes

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