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Michael S. Vogelius

Publications and source records attributed to Michael S. Vogelius.

13 recordsLinked to original sources
arXiv2026

On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof

This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $ω$ for which there exists a nontrivial incident $ω$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).

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arXiv2026

Regularity and non-degeneracy of $Φ^*[I]$ implies regularity of the fixed boundary $\partialΩ$

For a diffeomorphism $Φ$ of a domain $\overlineΩ$ onto itself, which is the identity on $\partialΩ$, we prove that local regularity of the push-forward $Φ^*[I]$ implies local regularity of $\partialΩ$, provided a certain non-degeneracy condition is satisfied. To be precise, if $ν$ is a normal to $\partialΩ$ at a point $P$, then the condition $(Φ^*[I](P)-I)ν\neq 0$ and the assumption that $Φ^*[I]$ is of class $C^{k+1,α}$ near $P$ imply that $\partialΩ$ is also of class $C^{k+1,α}$ near $P$. This result naturally complements recent regularity results for non-scattering inhomogeneities.

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arXiv2026

Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation

We investigate elliptic fractional equations in the whole space, involving zero order perturbations of the fractional Laplacian $(-Δ)^s$, $0 0$, obtained via contour integration and a limiting absorption principle. We show that its asymptotic behavior at infinity coincides with a rescaled version of the classical Helmholtz fundamental solution, thereby justifying the standard Sommerfeld radiation condition for compactly supported sources. In addition, using resolvent estimates and a limiting absorption framework, we establish existence and uniqueness of outgoing solutions for compactly supported data, and for weighted sources. We further derive a convolution representation of the solution in terms of the outgoing fundamental solution. For inhomogeneous media with compactly supported perturbations, we reformulate the problem as a Lippmann Schwinger integral equation of Fredholm type and prove unique solvability away from a discrete set of frequencies. Our analysis provides a rigorous foundation for scattering theory of fractional Helmholtz operators and offers a framework suitable for numerical implementation of these nonlocal wave propagation models.

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arXiv2026

Transmission Eigenvalues and Non-scattering

In this paper we survey some recent results concerning scattering and non-scattering in the context of the linear Helmholtz equation and inhomogeneities of nontrivial contrast. We examine isotropic as well as anisotropic media. Part of the survey deals with the so-called transmission spectrum, namely those wave numbers at which non-scattering potentially may occur. For wave numbers that are not transmission eigenvalues any incident wave leads to scattering, however, being at a transmission eigenvalue is far from su!cient to guarantee the occurence of non-scattering for even a single incident wave. For instance the inhomogeneity generically has to be smooth for non-scattering to occur. Similarly many smooth geometric shapes will be scattering for natural incident waves even at a transmission eigenvalue. Part of the survey discusses recent results of that nature.

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arXiv2025

Scattering of plane waves

We formulate a problem that can be viewed as a natural variation of the so-called Pompeiu or Schiffer problem in the context of scattering of plane waves for the Linear Helmholtz equation. For the two dimensional version of this variation, we establish conditions on the wave numbers and incident directions, that ensure a non-vanishing scattered field.

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arXiv2025

Scattering from analytic and piecewise analytic inhomogeneities

We study scattering for the linear Helmholtz operator in two dimensions and develop a technique, which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.

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arXiv2025

Perfectly matched layers in time domain. A simple two-dimensional error analysis

Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the two-dimensional wave equation and illustrate our conclusions with some numerical examples.

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arXiv2024

On the lack of external response of a nonlinear medium in the second-harmonic generation process

This paper concerns the scattering problem for a nonlinear medium of compact support, $D$, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency $ω$, generates additional waves at frequency $2ω$. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency $ω$ and $2ω$. We investigate whether there are situations in which the generated $2ω$ wave is localized inside $D$, that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in $D$ with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.

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arXiv2022

On the Regularity of Non-Scattering Anisotropic Inhomogeneities

In this paper we examine necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $C^{1,α}$ for some $0<α<1$.

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arXiv2021

Small perturbations in the type of boundary conditions for an elliptic operator

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $ω_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $ω_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $ω_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $ω_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $ω_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $ω_\varepsilon $ is a vanishing surfacic ball.

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arXiv2021

Singularities Almost Always Scatter: Regularity Results for Non-scattering Inhomogeneities

In this paper we examine necessary conditions for an inhomogeneity to be non-scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams [28] in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg [18], and Caffarelli [6].

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