0000000000052835

AUTHOR

Henrik Shahgholian

showing 5 related works from this author

Free boundary methods and non-scattering phenomena

2021

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…

FOS: Physical sciencesBoundary (topology)01 natural sciencesinversio-ongelmatTheoretical Computer ScienceMathematics - Analysis of PDEsMathematics (miscellaneous)ConverseFOS: MathematicsPoint (geometry)0101 mathematicsMathematical PhysicsComplement (set theory)MathematicsosittaisdifferentiaaliyhtälötQuadrature domainsScatteringApplied MathematicsResearch010102 general mathematicsMathematical analysisMathematical Physics (math-ph)010101 applied mathematicsComputational MathematicsObstacleInverse scattering problemAnalysis of PDEs (math.AP)Research in the Mathematical Sciences
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On the Porosity of Free Boundaries in Degenerate Variational Inequalities

2000

Abstract In this note we consider a certain degenerate variational problem with constraint identically zero. The exact growth of the solution near the free boundary is established. A consequence of this is that the free boundary is porous and therefore its Hausdorff dimension is less than N and hence it is of Lebesgue measure zero.

porosityLebesgue measureApplied MathematicsDegenerate energy levelsMathematical analysisZero (complex analysis)Boundary (topology)nonhomogeneous p-Laplace equationfree boundaryobstacle problemHausdorff dimensionVariational inequalityObstacle problemFree boundary problemAnalysisMathematicsJournal of Differential Equations
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Quadrature domains for the Helmholtz equation with applications to non-scattering phenomena

2022

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.

metaharmonic functionsmatematiikkapartial balayageyhtälötmean value theoremMathematics::Numerical Analysis35J05 35J15 35J20 35R30 35R35quadrature domainnon-scattering phenomenaMathematics - Analysis of PDEsFOS: MathematicsHelmholtz equationacoustic equationAnalysisAnalysis of PDEs (math.AP)
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A minimization problem with free boundary and its application to inverse scattering problems

2023

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.

Mathematics - Analysis of PDEsFOS: MathematicsAnalysis of PDEs (math.AP)35J05 35J15 35J20 35R30 35R35
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Free boundary methods and non-scattering phenomena

2021

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…

FOS: MathematicsFOS: Physical sciencesMathematical Physics (math-ph)Analysis of PDEs (math.AP)
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