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 $\Gamma_0$.