Unimprovability of estimates of H\"older constants for solutions of linear elliptic equations with measurable coefficients

M. V. Safonov

Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 269-281

Résumé

The author considers the validity of an estimate in the norm of the Hölder spaces $C^\beta$ for the solutions of linear elliptic equations $a_{ij}u_{x_ix_j}=0$, where $\nu|l|^2\leqslant a_{ij}l_il_j\leqslant\nu^{-1}|l|^2$ for all $l=(l_1,\dots,l_n)\in E_n$ ($n\geqslant2$, $\nu=\mathrm{const}>0$). This estimate does not depend on the smoothness of the coefficients $a_{ij}=a_{ij}(x)$. It is known (RZh. Mat., 1980, 6Б433) that such an estimate holds for sufficiently small exponents $\beta\in(0,1)$ depending on $n$ and $\nu$. In this paper it is proved that this dependence is essential: for every $\beta_0\in(0,1)$ one can exhibit a constant $\nu\in(0,1)$ and construct a sequence in $E_3$ of elliptic equations, of the indicated form with smooth coefficients, whose solutions converge uniformly in the unit ball to a function that does not belong to $C^{\beta_0}$. Bibliography: 5 titles.