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Eemeli Blåsten

Publications and source records attributed to Eemeli Blåsten.

13 recordsLinked to original sources
arXiv2018

Radiating and non-radiating sources in elasticity

In this work, we study the inverse source problem of a fixed frequency for the Navier's equation. We investigate that nonradiating external forces. If the support of such a force has a convex or non-convex corner or edge on their boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: The energy identity and new type exponential solutions for the Navier's equation.

math.AP↗
arXiv2018

Nonradiating sources and transmission eigenfunctions vanish at corners and edges

We consider the inverse source problem of a fixed wavenumber: study properties of an acoustic source based on a single far- or near-field measurement. We show that nonradiating sources having a convex or non-convex corner or edge on their boundary must vanish there. The same holds true for smooth enough transmission eigenfunctions. The proof is based on an energy identity from the enclosure method and the construction of a new type of planar complex geometrical optics solution whose logarithm is a branch of the square root. The latter allows us to deal with non-convex corners and edges.

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arXiv2017

On vanishing near corners of transmission eigenfunctions

Let $Ω$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $V\in L^\infty(Ω)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w\in L^2(Ω)$ and $k\in\mathbb{R}_+$, \[(Δ+k^2)v= 0 \quad \text{in } Ω,\] \[(Δ+k^2(1+V))w= 0 \quad \text{in } Ω,\] \[w-v \in H^2_0(Ω), \quad \lVert v \rVert_{L^2(Ω)}=1. \] We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{supp}(V)$. Indeed, it is proved that $v$ and $w$ vanish near a corner point on $\partial Ω$ in a generic situation where the corner possesses an interior angle less than $π$ and the potential function $V$ does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility.

math.AP↗
arXiv2017

Addendum to: "On vanishing near corners of transmission eigenfunctions"

In this addendum, we relax a restrictive assumption in [1] needed for the interior transmission eigenfunctions to hold the intrinsic geometric vanishing property in a corner. In addition we present in more detail another assumption which can also guarantee the vanishing property, namely being locally $H^2$ near the corner. This was mentioned briefly in [1].

math.AP↗
arXiv2017

Well-posedness of the Goursat problem and stability for point source inverse backscattering

We show logarithmic stability for the point source inverse backscattering problem under the assumption of angularly controlled potentials. Radial symmetry implies Hölder stability. Importantly, we also show that the point source equation is well-posed and also that the associated characteristic initial value problem, or Goursat problem, is well-posed. These latter results are difficult to find in the literature in the form required by the stability proof.

math.AP↗
arXiv2017

On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study

This paper is concerned with the intrinsic geometric structure of interior transmission eigenfunctions arising in wave scattering theory. We numerically show that the aforementioned geometric structure can be much delicate and intriguing. The major findings can be roughly summarized as follows. If there is a cusp on the support of the underlying potential function, then the interior transmission eigenfunction vanishes near the cusp if its interior angle is less than $π$, whereas the interior transmission eigenfunction localizes near the cusp if its interior angle is bigger than $π$. Furthermore, we show that the vanishing and blowup orders are inversely proportional to the interior angle of the cusp: the sharper the angle, the higher the convergence order. Our results are first of its type in the spectral theory for transmission eigenvalue problems, and the existing studies in the literature concentrate more on the intrinsic properties of the transmission eigenvalues instead of the transmission eigenfunctions. Due to the limitedness of the computing resources, our study is by no means exclusive and complete. We consider our study only in a certain geometric setup including corner, curved corner and edge singularities. Nevertheless, we believe that similar results hold for more general cusp singularities and rigorous theoretical justifications are much desirable. Our study enriches the spectral theory for transmission eigenvalue problems. We also discuss its implication to inverse scattering theory.

math.NA↗
arXiv2016

Translation-Invariant Estimates for Operators with Simple Characteristics

We prove $L^{2}$ estimates and solvability for a variety of simply characteristic constant coefficient partial differential equations $P(D)u=f$. These estimates \[||u||_{L^2(D_{r})}\le C\sqrt{d_{r}d_{s}} ||f||_{_{L^2(D_{s})}}\] depend on geometric quantities - the diameters $d_{r}$ and $d_{s}$ of the regions $D_{r}$, where we estimate $u$, and $D_{s}$, the support of $f$ - rather than weights. As these geometric quantities transform simply under translations, rotations, and dilations, the corresponding estimates share the same properties. In particular, this implies that they transform appropriately under change of units, and therefore are physically meaningful. The explicit dependence on the diameters implies the correct global growth estimates. The weighted $L^{2}$ estimates first proved by Agmon in order to construct the generalized eigenfunctions for Laplacian plus potential in $\mathbb{R}^{n}$, and the more general and precise Besov type estimates of Agmon and Hörmander, are all simple direct corollaries of the estimate above.

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arXiv2013

On the Gel'fand-Calderón inverse problem in two dimensions

We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schrödinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobolev embedding into the space of Hölder-continuous functions, by showing that a particular kind of space embeds into the space of continuous functions without any modulus of continuity. The inverse problem is considered in the second part of the thesis. We prove a new Carleman estimate for $\partial$. This estimate has a fast decay rate, which will allow us to consider potentials with very low regularity. After that we use Bukhgeim's oscillating exponential solutions, Alessandrini's identity and stationary phase to get information about the difference of the potentials from the difference of the Cauchy data. The stability estimate will be of logarithmic type, but works with potentials of low regularity.

math.AP↗
arXiv2013

Do corners always scatter?

We study time harmonic scattering for the Helmholtz equation in Rn. We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the relative scattering (a.k.a. far field) operator has a trivial kernel and cokernel at every real wavenumber.

math.AP↗
arXiv2011

Stability and uniqueness for the inverse problem of the Schrödinger equation in 2D with potentials in W^{ε,p}

This result will be published as part of my PhD thesis after some streamlining. This manuscript contains the proof of the claim, but is not peer-reviewed. We prove uniqueness and stability for the inverse problem of the 2D Schrödinger equation in the case that the potentials give well posed direct problems and are in W^{ε,p}(Ω), ε>0, p>2. The idea of the proof is to use Bukhgeim's oscillating exponential solutions. By Alessandrini's identity and stationary phase we get information about the difference of the potentials from the difference of the Dirichlet-Neumann maps. Using interpolation, we see that the the worst of the remainder terms decays with an exponent of 1 - ε - β. Here β is the exponent which we get in a norm estimate for the conjugated Cauchy operator. We get it arbitrarily close to 1, so there is uniqueness and stability when ε > 0.

math.AP↗
arXiv2011

The inverse problem of the Schrödinger equation in the plane; A dissection of Bukhgeim's result

The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schrödinger equation in the plane. The thesis is mainly based on Bukhgeim's paper and Kari Astala's seminar talk, which he gave the 11th and 18th September of 2008 at the University of Helsinki. Section 3 is devoted to the history and past results concerning some related problems: notably the inverse problem of the Schrödinger and conductivity equations in different settings. We also describe why some of the past methods do not work in the general case in a plane domain. Section 4 outlines Bukhgeim's result and sketches out the proof. This proof is a streamlined version of the one in Bukhgeim's paper with the stationary phase method based on Kari Astala's presentation. In the following section we prove all the needed lemmas which are combined in section 6 to prove the solvability of the inverse problem. The idea of the proof is simple. Given two Schrödinger equations with the same boundary data we get an orthogonality relation for the solutions of the two equations. Then we show the existence of certain oscillating solutions and insert these into the orthogonality relation. Then by a stationary phase argument we see that the two Schrödinger equations are the same. In the last section we contemplate an unclear detail in Bukhgeim's paper which Kari Astala pointed out in his seminar talk: without an extra argument Bukhgeim's proof shows the solvability of the inverse problem only for differentiable potentials instead of ones in L^p. But the special oscillating solutions exist even for L^p potentials.

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