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Thim, Johan
Publications (9 of 9) Show all publications
Thim, J. (2016). Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts. Annali di Matematica Pura ed Applicata, 195(2), 323-341
Open this publication in new window or tab >>Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts
2016 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 195, no 2, p. 323-341Article in journal (Refereed) Published
Abstract [en]

It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.

Place, publisher, year, edition, pages
SPRINGER HEIDELBERG, 2016
Keywords
Riesz potentials; Singular integrals; Weighted spaces; Radial functions; Spherical symmetry
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-127414 (URN)10.1007/s10231-014-0465-8 (DOI)000373086500003 ()
Available from: 2016-05-02 Created: 2016-04-26 Last updated: 2017-11-30
Kozlov, V. & Thim, J. (2016). Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators. Journal of Spectral Theory, 6(1), 99-135
Open this publication in new window or tab >>Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators
2016 (English)In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 6, no 1, p. 99-135Article in journal (Refereed) Published
Abstract [en]

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be C-1,C-alpha regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.

Place, publisher, year, edition, pages
EUROPEAN MATHEMATICAL SOC, 2016
Keywords
Hadamard formula; domain variation; asymptotics of eigenvalues; Neumann problem
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-129508 (URN)10.4171/JST/120 (DOI)000376418300005 ()
Available from: 2016-06-20 Created: 2016-06-20 Last updated: 2017-11-28
Thim, J. (2015). ASYMPTOTICS OF HADAMARD TYPE FOR EIGENVALUES OF THE NEUMANN PROBLEM ON C-1-DOMAINS FOR ELLIPTIC OPERATORS. Analysis & PDE, 8(7), 1695-1706
Open this publication in new window or tab >>ASYMPTOTICS OF HADAMARD TYPE FOR EIGENVALUES OF THE NEUMANN PROBLEM ON C-1-DOMAINS FOR ELLIPTIC OPERATORS
2015 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 8, no 7, p. 1695-1706Article in journal (Refereed) Published
Abstract [en]

This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C-1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C-1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C-1. Moreover, as an application we consider the case of a C-1-perturbation when the reference domain is of class C-1,C-alpha.

Place, publisher, year, edition, pages
MATHEMATICAL SCIENCE PUBL, 2015
Keywords
Hadamard formula; domain variation; asymptotics of eigenvalues; Neumann problem; C-1-domains
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123089 (URN)10.2140/apde.2015.8.1695 (DOI)000364339400005 ()
Available from: 2015-12-03 Created: 2015-12-03 Last updated: 2017-12-01
Thim, J. (2015). Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant. Potential Analysis, 43(1), 79-95
Open this publication in new window or tab >>Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 1, p. 79-95Article in journal (Refereed) Published
Abstract [en]

This article considers two weight estimates for the single layer potential - corresponding to the Laplace operator in R (N+1) - on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2015
Keywords
Single layer potentials; Lipschitz surface; Singular integrals; Weighted spaces; Homogeneous Sobolev spaces
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-120220 (URN)10.1007/s11118-015-9464-7 (DOI)000357132800004 ()
Available from: 2015-07-21 Created: 2015-07-20 Last updated: 2017-12-04
Kozlov, V., Thim, J. & Turesson, B.-O. (2014). Single layer potentials on surfaces with small Lipschitz constants. Journal of Mathematical Analysis and Applications, 418(2), 676-712
Open this publication in new window or tab >>Single layer potentials on surfaces with small Lipschitz constants
2014 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 418, no 2, p. 676-712Article in journal (Refereed) Published
Abstract [en]

This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in RN+1. Let Lambda(r) be the Lipschitz constant of phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L-p-spaces and local Sobolev spaces, where 1 less than p less than infinity, and results are presented in terms of Lambda. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Single layer potential; Lipschitz surface; Local estimates
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-108788 (URN)10.1016/j.jmaa.2014.04.013 (DOI)000336887700007 ()
Available from: 2014-07-07 Created: 2014-07-06 Last updated: 2017-12-05
Kozlov, V., Thim, J. & Turesson, B.-O. (2010). A Fixed Point Theorem in Locally Convex Spaces. Collectanea Mathematica (Universitat de Barcelona), 61(2), 223-239
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, E-ISSN 2038-4815, Vol. 61, no 2, p. 223-239Article in journal (Other academic) Published
Abstract [en]

For a locally convex space , where the topology is given by a familyof seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set . We require that there exists a linear and positive operator , acting on functions defined on the index set , such that for every

Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of . For a class of elements satisfying as , 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
Keywords
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: 2017-12-14Bibliographically approved
Thim, J., Kozlov, V. & Turesson, B.-O. (2009). Riesz Potential Equations in Local Lp-spaces.. Complex Variables and Elliptic Equations, 54(2), 125-151
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, E-ISSN 1747-6941, Vol. 54, no 2, p. 125-151Article in journal (Refereed) Published
Abstract [en]

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

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.

Keywords
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: 2017-12-14Bibliographically approved
Thim, J. (2009). Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach. (Doctoral dissertation). Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>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

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 , 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 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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2009. p. 12
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1235
Keywords
Singular integrals, Lipschitz surfaces
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-16280 (URN)978-91-7393-709-2 (ISBN)
Public defence
2009-02-20, Nobel BL32, B-huset, ingång 23, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2009-02-09 Created: 2009-01-12 Last updated: 2020-02-19Bibliographically approved
Thim, J. (2006). Riesz potentials and Riesz transforms in local Lp-spaces. (Licentiate dissertation). Linköping: Linköpings universitet
Open this publication in new window or tab >>Riesz potentials and Riesz transforms in local Lp-spaces
2006 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

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

The analysis is done in local Lp and Sobolev spaces, where the topologies are described by a family of semi-norms depending on a positive real parameter. Uniqueness and existence results are proved and asymptotic properties of solutions near the origin are established. Furthermore, we investigate properties of these operators in invariant subspaces.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2006. p. 68
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1276
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-153092 (URN)LiU-TEK-LIC-2006:57 (Local ID)9185643742 (ISBN)LiU-TEK-LIC-2006:57 (Archive number)LiU-TEK-LIC-2006:57 (OAI)
Presentation
2006-10-23, Glashuset, Hus B, ing. 25, Campus Valla, 13:15 (Swedish)
Opponent
Available from: 2019-01-07 Created: 2018-11-28 Last updated: 2023-02-15Bibliographically approved
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