On the oblique boundary problem with a stochastic inhomogeneity

  • We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.

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Metadaten
Author:Thomas Raskop, Martin Grothaus
URN (permanent link):urn:nbn:de:hbz:386-kluedo-14026
Serie (Series number):Schriften zur Funktionalanalysis und Geomathematik (25)
Document Type:Preprint
Language of publication:English
Year of Completion:2005
Year of Publication:2005
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2005/12/22
GND-Keyword:Galerkin-Methode ; Geodäsie ; Poisson-Gleichung ; Randwertproblem / Schiefe Ableitung ; Sobolev-Raum ; Stochastisches Feld
Source:Stochastics, Band 78(4), 2006, S. 233–257
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification (mathematics):35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J25 Boundary value problems for second-order elliptic equations
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Rxx Miscellaneous topics (For equations on manifolds, see 58Jxx; for manifolds of solutions, see 58Bxx; for stochastic PDE, see also 60H15) / 35R05 Partial differential equations with discontinuous coefficients or data
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Rxx Miscellaneous topics (For equations on manifolds, see 58Jxx; for manifolds of solutions, see 58Bxx; for stochastic PDE, see also 60H15) / 35R60 Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]
46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Exx Linear function spaces and their duals [See also 30H05, 32A38, 46F05] (For function algebras, see 46J10) / 46E35 Sobolev spaces and other spaces of \smooth" functions, embedding theorems, trace theorems
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G60 Random fields
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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