On non-local reflection for elliptic equations of the second order in $\mathbb{R}^{2}$ (the Dirichlet condition)

Abstract

Point-to-point reflection holding for harmonic functions subject to the Dirichlet or Neumann conditions on an analytic curve in the plane almost always fails for solutions to more general elliptic equations. We develop a non-local, point-to-compact set, formula for reflecting a solution of an analytic elliptic partial differential equation across a real-analytic curve on which it satisfies the Dirichlet conditions. We also discuss the special cases when the formula reduces to the pointto-point forms. 0 2000 Mathematics Subject Classification: Primary 35J15; Secondary 32D15.

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 [6,32,31,33]. (ii) A novel path argument for applying 'non-point-to-point' reflection principles to prove the analytical extension of wave fields in polygonal domains.…”

“…We refer to [6,9,11,18,25] for references related to harmonic functions. In [32,33] the reflection principle for the Helmholtz equation with the Dirichlet boundary condition given on a real analytic curve was derived, which was later extended to the Neumann boundary condition in [31]. In two dimensions, we believe that the reflection principle for the Helmholtz equation subject to the Robin boundary condition can be established analogously, following the approaches for harmonic functions [6] and the ideas in [32,33,31].…”

From reflections to a uniform elliptic growth

“…where c = λ 2 is a positive constant, c 0 (P, Γ) = 1, a = b = 0, and R = J 0 (λ (x − x 0 ) 2 + (y − y 0 ) 2 ). Thus, formula (4.30) could be slightly simplified, but it is still nonlocal [13].…”

“…Reflections are usually associated with continuation of boundary value problems. They were extensively studied by many researchers (see for example [2], [6], [8], [13], [15] and references therein). Analytic continuation of solutions to Cauchy problem for elliptic equations was studied in [10].…”

A reflection result for harmonic functions which vanish on a cylindrical surface

“…The special case of Theorem 2 where the harmonic function is of the form f ( x , x N ) follows easily from known reflection results in the plane (see Lewy [7]), since Δf + (N − 2)s −1 ∂f /∂s = 0 on the domain {(s, t) : |t| < a, φ(t) < s < 1} and f = 0 on the boundary line segment {1} × (−a, a). There are even explicit formulae for the extension in this case: see Savina [8]. (We are grateful to Dima Khavinson for these references.…”