The Schwarz reflection principle for harmonic functions in R2 subject to the Robin condition

Abstract

A non-local reflection formula for harmonic functions in R 2 satisfying the Robin boundary condition, α∂ n u +βu = 0, on a real-analytic curve is suggested. This formula generalizes the celebrated Schwarz reflection principle. It is also shown how the obtained formula reduces to well-known point-to-point reflection laws corresponding the Dirichlet and Neumann boundary conditions when one of the coefficients, α or β, vanishes.

Uniqueness to inverse acoustic scattering from coated polygonal obstacles with a single incoming wave

“…Helmholtz) equation subject to the Robin (resp. Dirichlet and Neumann) boundary condition(s) on a real-analytic subboundary [2,[34][35][36]. (ii) A novel path argument for applying 'non-point-to-point' reflection principles to prove the analytical extension of wave fields in polygonal domains.…”

Hölder stability estimate of Robin coefficient in corrosion detection with a single boundary measurement

“…Under such a condition, the solutions to the Laplace or elliptic equation with constant coefficients can be analytically extended into a neighboring (rectangular or annular) area of any sub-interval of G in 2 ⧹  W with a constant thickness. This is due to the global extension formula for flat and circular boundaries subject to the Robin boundary condition [6,15]. If G is a non-singular real-analytic curve, then a local extension is still possible only if the Robin coefficient is analytic; see [6].…”

“…This is due to the global extension formula for flat and circular boundaries subject to the Robin boundary condition [6,15]. If G is a non-singular real-analytic curve, then a local extension is still possible only if the Robin coefficient is analytic; see [6]. However, the extended area is determined by both the Schwarz function associated with G and the geometrical shape of Ω. Theorems 3.1 and 4.1 can be generalized to non-singular analytic curves, provided the thickness of the extended area keeps a positive lower bound from zero (this can be guaranteed, e.g.…”

“…n  Recently a more general reflection principle was established in [6] for harmonic functions subject to the Robin boundary condition with analytic coefficients in .…”

“…G Then the solution u to (1) can be analytically extended into the domain . S + In particular, when q x q 0 ( ) = is a constant, the extension formula is given explicitly by We refer to [6] for a proof performed for general non-singular real-analytic curves in 2  based on the idea of constructing reflected fundamental solutions (see [16]). The explicit extension formulae can also be directly justified following the argument in section 6.2 of this paper.…”