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 $u(x,y)\in\mathbb{R}^2$ 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, $\Gamma_0 :=\{y=0\}$, reflections are point to point when the given on $\Gamma_0$ conditions are $u=\partial_nu=0$, $u=\Delta u=0$ or $\partial_nu=\partial n\Delta u=0$, and point to a continuous set when $u=\partial_n\Delta u=0$ or $\partial_nu=\Delta u=0$ on $\Gamma_0$.