On an inversion theorem for conical regions in elasticity theory

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Two-dimensional stress singularities in wedges have already drawn attention since a long time. An inverse square-root stress singularity (in a 360° wedge) plays an important role in fracture mechanics. Recently some similar three-dimensional singularities in conical regions have been investigated, from which one may be also important in fracture mechanics. Spherical coordinates are r, θ, ϕ. The conical region occupied by the elastic homogeneous body (and possible anisotropic) has its vertex at r=0. The mantle of the cone is described by an arbitrary function f(θ, ϕ)=0. The displacement components be uξ. For special values of λ (eigenvalues) there exist states of displacements (eigenstates) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],which may satisfy rather arbitrary homogeneous boundary conditions along the generators. The paper brings a theorem which expresses that if λ is an eigenvalue, then also-1-λ is an eigenvalue. Though the theorem is related to a known theorem in Potential Theory (Kelvin's theorem), the proof has to be given along quite another line.