# A priori estimates in geometry and Sobolev spaces on open manifolds

Banach Center Publications (1992)

- Volume: 27, Issue: 1, page 141-146
- ISSN: 0137-6934

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topEichhorn, Jürgen. "A priori estimates in geometry and Sobolev spaces on open manifolds." Banach Center Publications 27.1 (1992): 141-146. <http://eudml.org/doc/262682>.

@article{Eichhorn1992,

abstract = {Introduction. For bounded domains in $R^n$ satisfying the cone condition there are many embedding and module structure theorem for Sobolev spaces which are of great importance in solving partial differential equations. Unfortunately, most of them are wrong on arbitrary unbounded domains or on open manifolds. On the other hand, just these theorems play a decisive role in foundations of nonlinear analysis on open manifolds and in solving partial differential equations. This was pointed out by the author in particular in [4]. But if the open Riemannian manifold $(M^n,g)$ and the considered Riemannian vector bundle (E,h) → M have bounded geometry of sufficiently high order then most of the Sobolev theorems can be preserved. The key for this are a priori estimates for the connection coefficients and the exponential map coming from curvature bounds. By means of uniform charts and trivializations and a uniform decomposition of unity the local euclidean arguments remain applicable. Only the compactness of embeddings is no more valid. This is the content of our main section 4.},

author = {Eichhorn, Jürgen},

journal = {Banach Center Publications},

keywords = {cone condition; embedding and module structure theorem for Sobolev spaces; partial differential equations; open Riemannian manifold; Riemannian vector bundle; Sobolev theorems},

language = {eng},

number = {1},

pages = {141-146},

title = {A priori estimates in geometry and Sobolev spaces on open manifolds},

url = {http://eudml.org/doc/262682},

volume = {27},

year = {1992},

}

TY - JOUR

AU - Eichhorn, Jürgen

TI - A priori estimates in geometry and Sobolev spaces on open manifolds

JO - Banach Center Publications

PY - 1992

VL - 27

IS - 1

SP - 141

EP - 146

AB - Introduction. For bounded domains in $R^n$ satisfying the cone condition there are many embedding and module structure theorem for Sobolev spaces which are of great importance in solving partial differential equations. Unfortunately, most of them are wrong on arbitrary unbounded domains or on open manifolds. On the other hand, just these theorems play a decisive role in foundations of nonlinear analysis on open manifolds and in solving partial differential equations. This was pointed out by the author in particular in [4]. But if the open Riemannian manifold $(M^n,g)$ and the considered Riemannian vector bundle (E,h) → M have bounded geometry of sufficiently high order then most of the Sobolev theorems can be preserved. The key for this are a priori estimates for the connection coefficients and the exponential map coming from curvature bounds. By means of uniform charts and trivializations and a uniform decomposition of unity the local euclidean arguments remain applicable. Only the compactness of embeddings is no more valid. This is the content of our main section 4.

LA - eng

KW - cone condition; embedding and module structure theorem for Sobolev spaces; partial differential equations; open Riemannian manifold; Riemannian vector bundle; Sobolev theorems

UR - http://eudml.org/doc/262682

ER -

## References

top- [1] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer, New York 1982. Zbl0512.53044
- [2] J. Dodziuk, Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Differential Geom. 16 (1981), 63-73. Zbl0456.58001
- [3] D. Ebin, Espace des métriques Riemanniennes et mouvement des fluides via les variétés d'applications, Lecture notes, Paris 1972. Zbl0242.58002
- [4] J. Eichhorn, Gauge theory on open manifolds of bounded geometry, Internat. J. Modern Physics, to appear. Zbl0890.58002
- [5] J. Eichhorn, The boundedness of connection coefficients and their derivatives, Math. Nachr. 152 (1991), 145-158. Zbl0736.53031
- [6] J. Eichhorn, Elliptic differential operators on noncompact manifolds, in: Teubner-Texte zur Math. 106, Teubner, 1988, 4-169.
- [7] R. Greene, Complete metrics of bounded curvature on noncompact manifolds, Arch. Math. (Basel) 31 (1978), 89-95. Zbl0373.53018

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