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Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This work is devoted to the equation

$\int_{S} \frac{u(y) \, dS(y)}{|x-y|^{N-1}} = f(x) \text{,} \qquad \qquad x \in S \text{,} \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)$

where S is the graph of a Lipschitz function φ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lp-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms.

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 Lp- 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 $\mathscr{X}$, where the topology is given by a family $\{p( \, \cdot \, ; \alpha )\}_{\alpha \in \Omega}$ of seminorms. We study the existence and uniqueness of fixed points for a mapping$\mathscr{K} \, : \; \mathscr{D_K} \rightarrow \mathscr{D_K}$ defined on a set $\mathscr{D_K} \subset \mathscr{X}$. 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 $u,v \in \mathscr{D_K}$,

$p(\mathscr{K}(u) - \mathscr{K}(v) \, ; \, \alpha ) \leq K(p(u-v \, ; \, \cdot \, )) (\alpha) \text{,} \qquad \qquad \alpha \in \Omega \text{.}$

Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p($\mathscr{K}(0)$ ; · ), we prove that there exists a fixed point of $\mathscr{K}$. For a class of elements satisfying Kn (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.

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 RN, 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.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1235
##### Keyword [en]
Singular integrals, Lipschitz surfaces
##### National Category
Mathematical Analysis
##### Identifiers
ISBN: 978-91-7393-709-2 (print)OAI: 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)
##### Supervisors
Available from: 2009-02-09 Created: 2009-01-12 Last updated: 2009-05-15Bibliographically approved
##### List of papers
1. Riesz Potential Equations in Local Lp-spaces.
Open this publication in new window or tab >>Riesz Potential Equations in Local Lp-spaces.
2009 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, Vol. 54, no 2, 125-151 p.Article in journal (Refereed) Published
##### Abstract [en]

We consider the following equation for the Riesz potential of order one:

$\int_{\mathbf{R}^N} \frac{u(y) \, dy}{|x-y|^{N-1}} = f(x) \text{,} \qquad x \in \mathbf{R}^N \text{.}$

Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove anexistence result and derive an asymptotic formula for solutions near the origin.Our analysis is carried out in local Lp-spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted Lp-spaces and homogenous Sobolev spaces.

##### Keyword
Riesz potentials, Singular integral operators
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16496 (URN)10.1080/17476930802669728 (DOI)
Available from: 2009-01-31 Created: 2009-01-29 Last updated: 2009-04-14Bibliographically approved
2. A Fixed Point Theorem in Locally Convex Spaces
Open this publication in new window or tab >>A Fixed Point Theorem in Locally Convex Spaces
2010 (English)In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, Vol. 61, no 2, 223-239 p.Article in journal (Other academic) Published
##### Abstract [en]

For a locally convex space $\mathscr{X}$, where the topology is given by a family$\{p( \, \cdot \, ; \alpha )\}_{\alpha \in \Omega}$of seminorms, we study the existence and uniqueness of fixed points for a mapping $\mathscr{K} : \, \mathscr{D_K} \rightarrow \mathscr{D_K}$ defined on some set $\mathscr{D_K} \subset \mathscr{X}$. We require that there exists a linear and positive operator $K$, acting on functions defined on the index set $\Omega$, such that for every $u,v \in \mathscr{D_K}$

$p(\mathscr{K}(u) - \mathscr{K}(v) \, ; \, \alpha ) \leq K(p(u-v \, ; \, \cdot \, )) (\alpha) \text{,} \qquad \qquad \alpha \in \Omega \text{.}$

Under some additional assumptions, one of which is the existence of a fixed point for the operator$K + p( \mathscr{K}(0) \, ; \, \cdot \,)$, we prove that there exists a fixed point of $\mathscr{K}$. For a class of elements satisfying$K^{n}(p(u \, ; \, \cdot \,)) (\alpha) \rightarrow 0$ as $n \rightarrow \infty$, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudo-differential equations with nonlinear terms.

##### Place, publisher, year, edition, pages
Universitat de Barcelona, 2010
##### Keyword
Fixed point theorem, Locally convex spaces, Ordinary differential equations, Pseudo-differential operators
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16537 (URN)10.1007/BF03191243 (DOI)000277332400006 ()
Available from: 2009-01-31 Created: 2009-01-30 Last updated: 2014-09-26Bibliographically approved
3. An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces
Open this publication in new window or tab >>An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces
##### Abstract [en]

This paper is devoted to the equation

$\int_{S} \frac{U(Q) \, dS(Q)}{|P-Q|^{N-1}} = F(P) \text{,} \qquad \qquad P \in S \text{,}$

where the surface S is the graph of a Lipschitz function φ on RN, which has a small Lipschitz constant. The integral in the left-hand side is the simple layer potential of the Laplacian in RN+1. Let Λ(r) be the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Our analysis is carried out in local Lp-spaces and local Sobolev spaces, where 1 < p < ∞, and results are presented in terms of Λ. Estimates of solutions to the equation for the single layer potential are provided, which can be used to obtain knowledge about behaviour of the solutions near a point on the surface. The estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Some examples are given when specific assumptions are placed on the function Λ.

##### Keyword
Lipschitz surfaces, Simple Layer potentials, Singular integral operators
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16538 (URN)
Available from: 2009-01-31 Created: 2009-01-30 Last updated: 2010-01-14Bibliographically approved
4. Invariant Properties of Riesz Potentials
Open this publication in new window or tab >>Invariant Properties of Riesz Potentials
##### Abstract [en]

In this paper, we collect some properties and estimates of Riesz potential operators, and also for the inverse on certain spaces of the Riesz potential operator of order one, 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.

##### Keyword
Riesz potentials, Riesz transforms, invariant subspaces, radial functions
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:liu:diva-16495 (URN)
Available from: 2009-01-31 Created: 2009-01-29 Last updated: 2010-01-14Bibliographically approved

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Thim, Johan

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