Spectral Problems Associated with Corner Singularities of by Kozlov V.A., Maz'ya V.G., Rossmann J.

By Kozlov V.A., Maz'ya V.G., Rossmann J.

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2) (n − 2) v dω = Ω a v dσ ∂Ω + for v ∈ ker A (0). 1, the function u0 = 1 is the eigenfunction corresponding to the eigenvalue λ = 0. The boundary value problem for the generalized eigenfunction u1 has the form −δu1 = n − 2 in Ω, ∂ν u1 + Q u1 = −a on ∂Ω. Therefore, (n − 2) v dω = − Ω v · δu1 dω = Ω (∂ν v · u1 − v · ∂ν u1 ) dω. 4. THE PROBLEM WITH OBLIQUE DERIVATIVE 51 By the boundary conditions on the functions v and u1 , the last integral is equal to v(Qu1 + a) − Q+ v · u1 dσ = ∂Ω av dσ. 2) is equivalent to the nonexistence of a generalized eigenfunction.

We derive the asymptotics of the function λ2 (S 2 \Eα ) for α → 2π. Let u(θ1 , θ2 ) be an eigenfunction to this eigenvalue. Then u(π −θ1 , θ2 ) is also an eigenfunction to this eigenvalue. Consequently, u(θ1 , θ2 ) = u(π−θ1 , θ2 ) or u(θ1 , θ2 ) = −u(π−θ1 , θ2 ). , λ2 (S 2 \Eα ) is an eigenvalue of the operator pencil to the Neumann problem for the half-sphere 2 S+ and u is an eigenfunction corresponding to this eigenvalue. Moreover, u dω = 0. 2 S+ 2 This implies λ2 (S 2 \Eα ) = λj (S+ ) for a certain index j ≥ 2.

4) u(π/2, θ2 ) = 0 for α < θ2 < 2π. 4). 1), we obtain λ2 (S 2 \Eα ) ∼ 1 2 | log(2π − α)| as α → 2π. 3. Finally, we consider the eigenvalues λj for a small domain on the unit sphere. Let G be a bounded domain in Rn−1 and Ωε = {ω ∈ S n−1 : ω /ε ∈ G}. Analogously to the case of the Dirichlet problem, there is the asymptotic representation λj (Ωε ) = µj (G) + O(1) ε for j ≥ 2, where µj (G) is the j-th eigenvalue of the Neumann problem for the Laplace operator in the domain G. 6. Applications to boundary value problems for the Laplace equation The results of this chapter lead to various assertions concerning regularity properties of solutions and solvability of boundary value problems for the Laplace equation in domains with singular boundary points, such as angular and conic vertices, edges.

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