Geodesic
Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients
Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 269-281 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SM_1988_60_1_a15,
     author = {M. V. Safonov},
     title = {Unimprovability of estimates of {H\"older} constants for solutions of linear elliptic equations with measurable coefficients},
     journal = {Sbornik. Mathematics},
     pages = {269--281},
     year = {1988},
     volume = {60},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_60_1_a15/}
}
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M. V. Safonov. Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. Sbornik. Mathematics, Tome 60 (1988) no. 1, pp. 269-281. http://geodesic.mathdoc.fr/item/SM_1988_60_1_a15/

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