Error estimates for interpolation by harmonic and biharmonic splines with annular geometry

The main result in this paper is an error estimate for interpolation by biharmonic splines in an annulus $\{x\in R^d:r_1\le \left|x\right|\le r_N\}$, with respect to a partition by concentric annular domains $\{x\in R^d:r_j<\left|x\right|<r_{j+1}\}$ for radii $0<r_1<....<r_N$. The biharmonic splines interpolate a smooth function on the spheres $\left|x\right|=r_j$ for $j=1,...,N$ and satisfy Dirichlet boundary conditions for $\left|x\right|=r_1$ and $\left|x\right|=r_N$. It is a remarkable fact that an elegant technique in the one-dimensional spline theory, namely a special orthogonality relation for the error, can be carried over to the case of biharmonic splines leading to an elegant proof. For the error estimates it is important to determine the smallest constant $c_d\left(\Omega \right)$ among all constants c satisfying$\underset{x\in \Omega }{\sup }\left|f\left(x\right)\right|\le c\underset{x\in \Omega }{\sup }\left|\Delta f\left(x\right)\right|$ for all $f\in C^2\left(\Omega \right)\cap C\left(\bar{\Omega }\right)$ vanishing on the boundary of the bounded domain Ω. In this paper we describe $c_d\left(\Omega \right)$ for an annulus $\Omega =A(r,R)$ and we will give the estimate$\min \{\frac{1}{2d},\frac{1}{8}\}{(R-r)}^2\le c_d\left(A(r,R)\right)\le \max \{\frac{1}{2d},\frac{1}{8}\}{(R-r)}^2$ where d is the dimension of the underlying space.