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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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
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:510 Mathematik
MSC-Classification (mathematics):35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
35J25 Boundary value problems for second-order elliptic equations
35R05 Partial differential equations with discontinuous coefficients or data
35R60 Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]
46E35 Sobolev spaces and other spaces of \smooth" functions, embedding theorems, trace theorems
60G60 Random fields

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