Geodesic
Asymptotics of the Solution of the Steklov Spectral Problem in a Domain with a Blunted Peak
Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 571-587

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We construct and justify the asymptotics of the eigenvalues and eigenfunctions of the Laplace equation with Steklov boundary conditions in a domain with an acute peak whose end of size $O(\varepsilon)$ is broken off. In particular, we establish that any positive eigenvalue with a fixed number turns out to be infinitesimal as $\varepsilon\to+0$ and the corresponding eigenfunction is localized in the $c\varepsilon$-neighborhood of the vertex of the peak.
Keywords: Steklov spectral problem, Laplace operator, domain with a blunted peak, Sobolev space, elliptic boundary-value problem, Neumann problem, Hardy inequality, Poincaré inequality, Green's formula, Hilbert space. 
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     author = {S. A. Nazarov},
     title = {Asymptotics of the {Solution} of the {Steklov} {Spectral} {Problem} in a {Domain} with a {Blunted} {Peak}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {571--587},
     publisher = {mathdoc},
     volume = {86},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_4_a9/}
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S. A. Nazarov. Asymptotics of the Solution of the Steklov Spectral Problem in a Domain with a Blunted Peak. Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 571-587. http://geodesic.mathdoc.fr/item/MZM_2009_86_4_a9/