On the dependence of the reflection operator on boundary conditions for biharmonic functions

Abstract

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x, y) ∈ R 2 subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures.In particular, in the special case of the boundary, Γ 0 := {y = 0}, reflections are point-topoint when the given on Γ 0 conditions are u = ∂ n u = 0, u = u = 0 or ∂ n u = ∂ n u = 0, and point to a continuous set when u = ∂ n u = 0 or ∂ n u = u = 0 on Γ 0 .

Integral equations for biharmonic data completion

“…The element v is zero also in a region containing Γ 2 in its interior (for an explicit extension formula for the biharmonic equation with zero Dirichlet and Neumann condition, see [13,Theorem 1] and [23,Theorem 3.2]; see further [1] and [14]). We now apply the normal derivative N y in (38).…”

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

“…G depends on the boundary condition as well, but the later is beyond the scope of this paper, see[2],[25] and[27] for some relevant results.…”