M. S. Agranovich, B. A. Amosov

On Fourier Series In Eigenfunctions Of Elliptic
Boundary Value Problems

We consider a general elliptic formally self-adjoint problem in a bounded domain $\Omega\subset\mathbb{R}^n$ with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in $L_2(\Omega)$ corresponding to this problem has an orthonormal basis $\{u_l\}$ of eigenfunctions, which are infinitely smooth in $\overline\Omega$. However, the system $\{u_l\}$ is not a basis in Sobolev spaces $H^t(\Omega)$ of high order.

We note and discuss the following possibility: for an arbitrarily large $t$, for each function $u\in H^t(\Omega)$ one can explicitly construct a function $u_0\in H^t(\Omega)$ such that the Fourier series of the difference $u-u_0$ in the functions $u_l$ converges to this difference in $H^t(\Omega)$. Moreover, the function $u(x)$ is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for $u$ are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of $u-u_0$. The function $u_0$ is obtained by applying some linear operator to these right-hand sides.