On the dependence of the reflection operator on boundary conditions for biharmonic functions
Tatiana V. Savina · arXiv (Cornell University) · 2010
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, $Γ_0 :=\{y=0\}$, reflections are point to point when the given on $Γ_0$ conditions are $u=\partial_nu=0$, $u=Δu=0$ or $\partial_nu=\partial nΔu=0$, and point to a continuous set when $u=\partial_nΔu=0$ or $\partial_nu=Δu=0$ on $Γ_0$.