Boundary value problems for a class of elliptic operator pencils

Denk, Robert; Mennicken, Reinhard; Волевич, Л Р

doi:10.1007/bf01228606

Cited by 13 publications

(26 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely, we have the following result. This result is essentially contained in [1], and in the present formulation it is proved in [13]. For completeness, we sketch the proof here.…”

Section: A(ξ D T ε)V(t)mentioning

confidence: 93%

“…In [1] additional restrictions of a technical nature concerning the simplicity of the roots were used. The paper [13] contains a proof that does not use these restrictions; this proof is basically reproduced in the proofs of Propositions 1.5 and 1.7 of the present paper. A new feature of the present proof is that the regularity condition in [1] is a consequence of the condition of proper small parameter ellipticity.…”

Section: 3mentioning

confidence: 94%

“…The condition of weak parameter ellipticity was introduced in [13,14]. For µ = 0 it becomes the Agranovich-Vishik parameter-ellipticity condition.…”

Section: A(ξ D T ε)V(t)mentioning

confidence: 99%

“…In the case of boundary conditions independent of ε, the problems (0.13), (0.14) were studied in [13,14].…”

Section: A(ξ D T ε)V(t)mentioning

confidence: 99%

See 3 more Smart Citations

The Vishik–Lyusternik method in elliptic problems with a small parameter

Волевич

2006

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

Abstract. We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the existence of a priori estimates that are uniform with respect to the parameter. The proof of uniform estimates is based on the construction of the exponential boundary layer introduced in the classical paper by Vishik and Lyusternik.

show abstract

“…More precisely, we have the following result. This result is essentially contained in [1], and in the present formulation it is proved in [13]. For completeness, we sketch the proof here.…”

Section: A(ξ D T ε)V(t)mentioning

confidence: 93%

Section: 3mentioning

confidence: 94%

See 2 more Smart Citations

The Vishik–Lyusternik method in elliptic problems with a small parameter

Волевич

2006

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The definition of the "shifted" function has a "geometrical" interpretation (see [5]). Suppose Γ is a convex polygon in R 2 (p,q) with vertices (p j , q j ), j = 0, .…”

Section: Resultsmentioning

confidence: 99%

Elliptic boundary value problems with large parameter for mixed order systems

Denk

Volevič²

2002

Partial Differential Equations

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper boundary value problems are studied for systems with large parameter, elliptic in the sense of Douglis-Nirenberg. We restrict ourselves on model problems acting in the half-space. It is possible to define parameter-ellipticity for such problems, in particular we formulate ShapiroLopatinskii type conditions on the boundary operators. It can be shown that parameter-elliptic boundary value problems are uniquely solvable and that their solutions satisfy uniform two-sided a priori estimates in parameterdependent norms. We essentially use ideas from Newton's polygon method and of Vishik-Lyusternik boundary layer theory.

show abstract

On elliptic operator pencils with general boundary conditions

Denk

Mennicken

Волевич

2001

Integr equ oper theory

Self Cite

View full text Add to dashboard Cite

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro-Lopatinskii condition. It is shown that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates for problems of Vishik-Lyusternik type containing a small parameter.

show abstract

Boundary value problems for a class of elliptic operator pencils

Cited by 13 publications

References 15 publications

The Vishik–Lyusternik method in elliptic problems with a small parameter

The Vishik–Lyusternik method in elliptic problems with a small parameter

Elliptic boundary value problems with large parameter for mixed order systems

On elliptic operator pencils with general boundary conditions

Contact Info

Product

Resources

About