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 
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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     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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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[4] Cordes H. O., “Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen”, Math. Ann., 131 (1956), 278–312 ; Kordes G. S., “O pervoi kraevoi zadache dlya kvazilineinykh differentsialnykh uravnenii vtorogo poryadka bolee chem s dvumya peremennymi”, Matematika (sb. perevodov), 3:2 (1959), 75–107 | DOI | MR | Zbl

[5] Nadirashvili N. S., “K zadache s naklonnoi proizvodnoi”, Matem. sb., 127(169) (1985), 398–416 | MR