Smooth coefficients and boundary make elliptic eigenfunctions smooth

Bootstrap. Since $u\in H_0^1(\mathrm{\Omega })\subset L^2(\mathrm{\Omega })$, the right-hand side $\lambda u$ lies in $L^2(\mathrm{\Omega })$; the $k=0$ case of [F2] gives $u\in H^2(\mathrm{\Omega })$. Then $\lambda u\in H^2(\mathrm{\Omega })$, and the $k=2$ case gives $u\in H^4(\mathrm{\Omega })$; iterating, $u\in H^{2j}(\mathrm{\Omega })$ for every $j$, hence $u\in H^m(\mathrm{\Omega })$ for every $m$.

Smooth representative and boundary values. All Sobolev orders are available by step 1.1. Regard $(L-\lambda )u=0$ as a zero-trace weak Dirichlet problem for the operator whose principal and first-order coefficients are those of $L$ and whose zeroth-order coefficient is $c-\lambda$. These coefficients remain smooth and uniformly elliptic. Apply [F3] to this operator with the smooth datum $0$; it gives a representative $\tilde{u}\in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$ satisfying $(L-\lambda )\tilde{u}=0$ pointwise, equivalently $L\tilde{u}=\lambda \tilde{u}$, with $\tilde{u}{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$.

Conclusion. The eigenfunction of a symmetric uniformly elliptic operator with smooth coefficients on a bounded $C^{\mathrm{\infty }}$ domain is smooth up to the boundary and satisfies the eigen-equation pointwise with zero boundary values; the spectral construction itself needs only the weak eigenpair, and this corollary records the regularity supplied by the estimates of this page.