Andreas Rosén - Academia.edu (original) (raw)
Papers by Andreas Rosén
HAL (Le Centre pour la Communication Scientifique Directe), 2012
We continue the development, by reduction to a first order system for the conormal gradient, of L... more We continue the development, by reduction to a first order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.
Computers & Mathematics with Applications
An integral equation reformulation of the Maxwell transmission problem is presented. The reformul... more An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus 0 and 1. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus 1 surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the more well-known Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented 13-digit accuracy both for transmitted and scattered fields.
Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators... more Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in A 2. The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah-Patodi-Singer boundary conditions. Contents 1. Introduction 2.1. Coverings 2.2. Weights and operators 2.3. Functional calculus 3. Proof of quadratic estimates 3.1. Localization on M 3.2. Pullback from M to R n 3.3. Extension of weights and coefficients 3.4. Estimates on R n 4. Degenerate boundary value problems 4.1. The boundary DB operator 4.2. Atiyah-Patodi-Singer conditions 4.3. Neumann and Dirichlet conditions References
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, ... more We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the L 1 Carleson functional and L q (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introdu... more As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L 2-Whitney averages belongs to L 2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X , and characterize the pointwise multipliers from X to L 2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.
Geometric Multivector Analysis, 2019
Chapter 11 should contain the material from differential geometry needed to read the present chap... more Chapter 11 should contain the material from differential geometry needed to read the present chapter. Section 12.1 builds on part of Chapter 10.
Geometric Multivector Analysis, 2019
This chapter builds on Chapter 3. The reader is assumed to know the spectral theorem for normal o... more This chapter builds on Chapter 3. The reader is assumed to know the spectral theorem for normal operators from linear algebra, as well as basic plane complex geometry. Some basic knowledge of Lie groups and algebras, as well as special relativity, is helpful.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
This chapter builds on Chapters 3 and 4, and uses the material in Sections 1.4 and 1.5. Any knowl... more This chapter builds on Chapters 3 and 4, and uses the material in Sections 1.4 and 1.5. Any knowledge of representation theory is helpful, but the presentation is self-contained and should be accessible to anyone with a solid background in linear algebra.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
This chapter is not where to start reading this book, which rather is Chapter 2. The material in ... more This chapter is not where to start reading this book, which rather is Chapter 2. The material in the present chapter is meant to be used as a reference for some background material and ideas from linear algebra, which are essential to this book, in particular to the first part of it on algebra and geometry consisting of Chapters 2 through 5.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
The reader should be familiar with the basic ideas of differential geometry. Section 11.1 gives a... more The reader should be familiar with the basic ideas of differential geometry. Section 11.1 gives a short survey of the required material and fixes notation. Section 11.2 builds on Chapter 7, and Section 11.6 builds on Chapter 5. Section 11.4 uses Section 4.5.
Advances in Computational Mathematics, 2021
Two recently derived integral equations for the Maxwell transmission problem are compared through... more Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.
Integral Equations and Operator Theory, 2021
A new integral equation formulation is presented for the Maxwell transmission problem in Lipschit... more A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.
Journal of the European Mathematical Society, 2018
We prove continuity and surjectivity of the trace map onto L p (R n), from a space of functions o... more We prove continuity and surjectivity of the trace map onto L p (R n), from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the BMO case, related to the Corona theorem. We also prove L p Carleson approximability results for solutions to elliptic non-smooth divergence form equations, which generalize results in the case p = ∞ by Hofmann, Kenig, Mayboroda and Pipher.
Applicable Analysis, 2016
We present a new integral equation for solving the Maxwell scattering problem against a perfect c... more We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two by two matrices as values, using a spin representation of the fields.
HAL (Le Centre pour la Communication Scientifique Directe), 2012
We continue the development, by reduction to a first order system for the conormal gradient, of L... more We continue the development, by reduction to a first order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.
Computers & Mathematics with Applications
An integral equation reformulation of the Maxwell transmission problem is presented. The reformul... more An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus 0 and 1. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus 1 surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the more well-known Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented 13-digit accuracy both for transmitted and scattered fields.
Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators... more Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in A 2. The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah-Patodi-Singer boundary conditions. Contents 1. Introduction 2.1. Coverings 2.2. Weights and operators 2.3. Functional calculus 3. Proof of quadratic estimates 3.1. Localization on M 3.2. Pullback from M to R n 3.3. Extension of weights and coefficients 3.4. Estimates on R n 4. Degenerate boundary value problems 4.1. The boundary DB operator 4.2. Atiyah-Patodi-Singer conditions 4.3. Neumann and Dirichlet conditions References
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, ... more We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the L 1 Carleson functional and L q (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introdu... more As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L 2-Whitney averages belongs to L 2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X , and characterize the pointwise multipliers from X to L 2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.
Geometric Multivector Analysis, 2019
Chapter 11 should contain the material from differential geometry needed to read the present chap... more Chapter 11 should contain the material from differential geometry needed to read the present chapter. Section 12.1 builds on part of Chapter 10.
Geometric Multivector Analysis, 2019
This chapter builds on Chapter 3. The reader is assumed to know the spectral theorem for normal o... more This chapter builds on Chapter 3. The reader is assumed to know the spectral theorem for normal operators from linear algebra, as well as basic plane complex geometry. Some basic knowledge of Lie groups and algebras, as well as special relativity, is helpful.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
This chapter builds on Chapters 3 and 4, and uses the material in Sections 1.4 and 1.5. Any knowl... more This chapter builds on Chapters 3 and 4, and uses the material in Sections 1.4 and 1.5. Any knowledge of representation theory is helpful, but the presentation is self-contained and should be accessible to anyone with a solid background in linear algebra.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
This chapter is not where to start reading this book, which rather is Chapter 2. The material in ... more This chapter is not where to start reading this book, which rather is Chapter 2. The material in the present chapter is meant to be used as a reference for some background material and ideas from linear algebra, which are essential to this book, in particular to the first part of it on algebra and geometry consisting of Chapters 2 through 5.
Geometric Multivector Analysis, 2019
Geometric Multivector Analysis, 2019
The reader should be familiar with the basic ideas of differential geometry. Section 11.1 gives a... more The reader should be familiar with the basic ideas of differential geometry. Section 11.1 gives a short survey of the required material and fixes notation. Section 11.2 builds on Chapter 7, and Section 11.6 builds on Chapter 5. Section 11.4 uses Section 4.5.
Advances in Computational Mathematics, 2021
Two recently derived integral equations for the Maxwell transmission problem are compared through... more Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.
Integral Equations and Operator Theory, 2021
A new integral equation formulation is presented for the Maxwell transmission problem in Lipschit... more A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.
Journal of the European Mathematical Society, 2018
We prove continuity and surjectivity of the trace map onto L p (R n), from a space of functions o... more We prove continuity and surjectivity of the trace map onto L p (R n), from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the BMO case, related to the Corona theorem. We also prove L p Carleson approximability results for solutions to elliptic non-smooth divergence form equations, which generalize results in the case p = ∞ by Hofmann, Kenig, Mayboroda and Pipher.
Applicable Analysis, 2016
We present a new integral equation for solving the Maxwell scattering problem against a perfect c... more We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two by two matrices as values, using a spin representation of the fields.