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Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic ApproachPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Linköping: Linköping University Electronic Press , 2009. , 12 p.
##### Series

Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1235
##### Keyword [en]

Singular integrals, Lipschitz surfaces
##### National Category

Mathematical Analysis
##### Identifiers

URN: urn:nbn:se:liu:diva-16280ISBN: 978-91-7393-709-2OAI: oai:DiVA.org:liu-16280DiVA: diva2:158270
##### Public defence

2009-02-20, Nobel BL32, B-huset, ingång 23, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
##### Opponent

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#####

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Available from: 2009-02-09 Created: 2009-01-12 Last updated: 2009-05-15Bibliographically approved
##### List of papers

This work is devoted to the equation

where *S* is the graph of a Lipschitz function φ on **R ^{N}** with small Lipschitz constant, and

In Paper 1, we consider the case when *S* is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted *L ^{p}*- and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces.

In Paper 2, we present a fixed point theorem for a locally convex space , where the topology is given by a family of seminorms. We study the existence and uniqueness of fixed points for a mapping defined on a set . It is assumed that there exists a linear and positive operator *K*, acting on functions defined on the index set Ω, such that for every ,

Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p( ; · ), we prove that there exists a fixed point of . For a class of elements satisfying *K ^{n}* (p(u ; · ))(α) → 0 as

In Paper 3, we treat equation (1) in the case when *S* is a general Lipschitz surface and 1 < *p* < ∞. Our results are presented in terms of Λ(*r*), which is the Lipschitz constant of φ on the ball centered at the origin with radius *2r*. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on *S* in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2.

In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on **R ^{N}**, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.

1. Riesz Potential Equations in Local L^{p}-spaces.$(function(){PrimeFaces.cw("OverlayPanel","overlay158259",{id:"formSmash:j_idt412:0:j_idt416",widgetVar:"overlay158259",target:"formSmash:j_idt412:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. A Fixed Point Theorem in Locally Convex Spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay158260",{id:"formSmash:j_idt412:1:j_idt416",widgetVar:"overlay158260",target:"formSmash:j_idt412:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces$(function(){PrimeFaces.cw("OverlayPanel","overlay158261",{id:"formSmash:j_idt412:2:j_idt416",widgetVar:"overlay158261",target:"formSmash:j_idt412:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Invariant Properties of Riesz Potentials$(function(){PrimeFaces.cw("OverlayPanel","overlay158264",{id:"formSmash:j_idt412:3:j_idt416",widgetVar:"overlay158264",target:"formSmash:j_idt412:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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