The outer oblique boundary problem of potential theory

In this article we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in [1]. Moreover we prove regularisation results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkinm approximation. The results are applicable to problems from Geomathematics, see e.g. [2] and [3]

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

Limit Formulae and Jump Relations of Potential Theory in Sobolev Spaces

In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from [14]. Additionally, we need a result about the limit formulae in L2-norm, which can be found in [16], and a reduction result which we took from [19]. Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. [11], from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also

Die Analysis der schiefachsiger Randwertprobleme und der Grenzwertgleichungen motiviert durch geomathematische Probleme

This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic functions. Moreover we prove the convergence in the Hölder spaces. Finally we give an application of the limit formulae and jump relations. We generalize a known density of several function systems from Geomathematics in the Lebesgue spaces of square integrable measureable functions, to density in Sobolev spaces, based on the results proved before. Therefore we have prove the limit formula of the single layer potential in dual spaces of Soboelv spaces, where also the layer function is an element of such a distribution space.In dieser Disseration werden zwei Themen behandelt. Beide stehen in engem Zusammenhang mit Randwertproblem für die Poisson Gleichung bzw. der Laplace Gleichung. Zum einen untersuchen wir schiefachsige Randwertprobleme für die Poisson Gleichung, zum anderen behandeln wir die sogenannten Grenzwertgleichungen der Potentialtheorie. Dementsprechend unterteilen wir diese Zusammenfassung. Wir beginnen mit den schiefachsigen Randwertproblemen. Hier beweisen wir Existenz- und Eindeutigkeitsaussagen für Lösungen des äußeren Randwertproblems unter sehr schwachen Anforderungen an Gebiet, Koeffizienten und Inhomogeitäten. Hauptwerkzeuge sind die Kelvin Transformation, sowie der Lösungsoperator für das innere Problem, welcher bereits in meiner Diplomarbeit zur Verfügung gestellt wurde. Wir beweisen ein Regularisierungsresultat für die schwache Lösung des inneren und äußeren Problems. Zudem untersuchen wir die nicht zulässigen Richtungen für das schiefachsige Vektorfeld, beweisen ein Resultat f¨ur stochastische Inhomogenitäten und implementieren eine Ritz-Galerkin-Approximations Methode. Schließlich zeigen wir noch das unsere Resultate auf konkrete Probleme der Geomathematik anwendbar sind. Kommen wir nun zu den Grenzwertgleichungen der Potentialtheorie. Hierbei kombinieren wir die bekannte Theorie mit dem modernen Konzept der Sobolev Räume. Die Konvergenz in den Lebesgue Räumen der quadratisch integrierbaren messbaren Funktionen ist bereits in der Literatur bewiesen. Unsere Verallgemeinerung besteht in der Konvergenz der schwachen Ableitungen in dieser Norm, d.h. in den sogenannten Sobolev Räumen, wobei dann auch die Belegungen der potentiale Elemente von Sobolev Räumen sind und die Fläche eine ausreichend glatte Untermannigfaltigkeit im dreidimensionalen Raum ist. Als erstes untersuchen wir die Konvergenz tangentialer Ableitungen des Einschicht-Potentials, des Zweischicht-Potentials, sowie deren Normalenableitungen erster Ordnung in Supremums-Norm. Für den Beweis benutzen wir ein bekanntes Resultat über die Regularität der genannten Potentiale. Darüber hinaus können wir zeigen das die Konvergenz sogar in den Hölder Räumen hält. Diese Resultate, zusammen mit der Konvergenz in den Lebesgue Räumen sind die Hauptzutaten für den Beweis in den Sobolev Normen. Außerdem benutzen wir noch eine Reduktionsformel für die Normalenableitungen harmonischer Funktionen. Schließlich wenden wir die bewiesenen Resultate an, indem wir zeigen das bestimmte Funktionensysteme der Geomathematik dicht in den Sobolev Räumen sind. Dazu beweisen wir unter anderem auch die Grenzwertgleichung für das Einschicht-Potential in den Dualräumen der Sobolev Räume für Belegungen ebenfalls aus diesen Dualräumen

Die Analysis der schiefachsiger Randwertprobleme und der Grenzwertgleichungen motiviert durch geomathematische Probleme

This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic functions. Moreover we prove the convergence in the Hölder spaces. Finally we give an application of the limit formulae and jump relations. We generalize a known density of several function systems from Geomathematics in the Lebesgue spaces of square integrable measureable functions, to density in Sobolev spaces, based on the results proved before. Therefore we have prove the limit formula of the single layer potential in dual spaces of Soboelv spaces, where also the layer function is an element of such a distribution space.In dieser Disseration werden zwei Themen behandelt. Beide stehen in engem Zusammenhang mit Randwertproblem für die Poisson Gleichung bzw. der Laplace Gleichung. Zum einen untersuchen wir schiefachsige Randwertprobleme für die Poisson Gleichung, zum anderen behandeln wir die sogenannten Grenzwertgleichungen der Potentialtheorie. Dementsprechend unterteilen wir diese Zusammenfassung. Wir beginnen mit den schiefachsigen Randwertproblemen. Hier beweisen wir Existenz- und Eindeutigkeitsaussagen für Lösungen des äußeren Randwertproblems unter sehr schwachen Anforderungen an Gebiet, Koeffizienten und Inhomogeitäten. Hauptwerkzeuge sind die Kelvin Transformation, sowie der Lösungsoperator für das innere Problem, welcher bereits in meiner Diplomarbeit zur Verfügung gestellt wurde. Wir beweisen ein Regularisierungsresultat für die schwache Lösung des inneren und äußeren Problems. Zudem untersuchen wir die nicht zulässigen Richtungen für das schiefachsige Vektorfeld, beweisen ein Resultat f¨ur stochastische Inhomogenitäten und implementieren eine Ritz-Galerkin-Approximations Methode. Schließlich zeigen wir noch das unsere Resultate auf konkrete Probleme der Geomathematik anwendbar sind. Kommen wir nun zu den Grenzwertgleichungen der Potentialtheorie. Hierbei kombinieren wir die bekannte Theorie mit dem modernen Konzept der Sobolev Räume. Die Konvergenz in den Lebesgue Räumen der quadratisch integrierbaren messbaren Funktionen ist bereits in der Literatur bewiesen. Unsere Verallgemeinerung besteht in der Konvergenz der schwachen Ableitungen in dieser Norm, d.h. in den sogenannten Sobolev Räumen, wobei dann auch die Belegungen der potentiale Elemente von Sobolev Räumen sind und die Fläche eine ausreichend glatte Untermannigfaltigkeit im dreidimensionalen Raum ist. Als erstes untersuchen wir die Konvergenz tangentialer Ableitungen des Einschicht-Potentials, des Zweischicht-Potentials, sowie deren Normalenableitungen erster Ordnung in Supremums-Norm. Für den Beweis benutzen wir ein bekanntes Resultat über die Regularität der genannten Potentiale. Darüber hinaus können wir zeigen das die Konvergenz sogar in den Hölder Räumen hält. Diese Resultate, zusammen mit der Konvergenz in den Lebesgue Räumen sind die Hauptzutaten für den Beweis in den Sobolev Normen. Außerdem benutzen wir noch eine Reduktionsformel für die Normalenableitungen harmonischer Funktionen. Schließlich wenden wir die bewiesenen Resultate an, indem wir zeigen das bestimmte Funktionensysteme der Geomathematik dicht in den Sobolev Räumen sind. Dazu beweisen wir unter anderem auch die Grenzwertgleichung für das Einschicht-Potential in den Dualräumen der Sobolev Räume für Belegungen ebenfalls aus diesen Dualräumen

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

Limit Formulae and Jump Relations of Potential Theory in Sobolev Spaces

In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from [14]. Additionally, we need a result about the limit formulae in L2-norm, which can be found in [16], and a reduction result which we took from [19]. Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. [11], from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also

The outer oblique boundary problem of potential theory

In this article we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in [1]. Moreover we prove regularisation results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkinm approximation. The results are applicable to problems from Geomathematics, see e.g. [2] and [3]