| 1. |
- Shahgholian, Henrik, et al.
(author)
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The obstacle problem with singular coefficients near Dirichlet data
- 2017
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : Elsevier. - 0294-1449 .- 1873-1430. ; 34:2, s. 293-334
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Journal article (peer-reviewed)abstract
- In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary B boolean AND {x1 = 0} in the obstacle type problem {div(x(1)(a) del u) = X-{u>0} in B+, u=0 on B boolean AND {x(1) = 0} where a < 1, B+ = B boolean AND {x(1) > 0}, B is the unit ball in R-n and n > 2 is an integer. Let Gamma = B+ boolean AND partial derivative{u > 0} be the free boundary and assume that the origin is a contact point, i.e. 0 epsilon (Gamma) over bar. We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a C-1 function and there is a uniform modulus of continuity for the derivatives of this function.
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| 2. |
- Helsing, Johan, et al.
(author)
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Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance
- 2017
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In: Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 34:4, s. 991-1011
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Journal article (peer-reviewed)abstract
- We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.
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