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- W2016045008 abstract "The Dirichlet problem for the region of the plane inside closed smooth curve C for second-order elliptic equations is considered. It is shown that under certain circumstances the solution u can be written uniquely in the form u(P) = ∝cF(P, Q) g(Q) dsQ, where F(P, Q) is the fundamental solution of the elliptic equation, and g ϵ L2 if the boundary value function f is absolutely continuous with square integrable derivative (f ϵ W); and u(P) = p(F(P, ·)) where p is a unique bounded linear functional on W if f ϵ L2. These representations are valid in the exterior of C also. As special cases with slight modifications, the exterior Dirichlet problems for the Helmholtz and Laplace equations are mentioned. It is shown also that if kernel F(P′, Q), with P′ and Q on C, has a complete set of eigenfunctions {ψk(P′)} then u(P) can be expanded in a series of their extensions {ψk(P)}, where ψk(P) = λk ∝cF(P, Q) ψk(Q) dsQ." @default.
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- W2016045008 date "1976-05-01" @default.
- W2016045008 modified "2023-10-14" @default.
- W2016045008 title "Some representations of solutions of elliptic equations" @default.
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- W2016045008 doi "https://doi.org/10.1016/0022-247x(76)90217-1" @default.
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