On the regularity of eigenfunctions of the Laplace operator on a Lipschitz manifold (Q1098466)

Let M be a compact Lipschitz n-manifold and g a metric with \(L_{\infty}\) coefficients and ``admissible'', i.e. compatible with the changes of charts and such that for every r-form \(\omega\) \((0\leq r\leq n)\) \[ k\int _{M}\omega \wedge^*_ E \omega \leq \int _{M}\omega \wedge^*_ g \omega \leq K \int _{M}\omega \wedge^*_ E \omega \] holds, where \({\;}^*_ E\) (resp. \({\;}^*_ g)\) is the Hodge star operator of the Euclidean metric (resp. of the metric g) [\textit{N. Teleman}, Publ. Math., Inst. Hautes Etud. Sci. 58, 251-290 (1983; Zbl 0531.58044)]. The authors prove that the eigenfunctions of the Laplace-Beltrami operator (with respect to g) belong to \(H_{1,2+\epsilon}(M)\cap C^{0,s}(M)\) for some \(\epsilon >0\) and \(s>0\). If \(n>1\), the indices \(\epsilon\) and s depend only on K/k and that in an essential way, as is shown by means of an example.