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This article is cited in 13 scientific papers (total in 13 papers)
Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients
M. V. Safonov
Abstract:
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.
Received: 17.09.1985
Citation:
M. V. Safonov, “Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients”, Math. USSR-Sb., 60:1 (1988), 269–281
Linking options:
https://www.mathnet.ru/eng/sm1780https://doi.org/10.1070/SM1988v060n01ABEH003167 https://www.mathnet.ru/eng/sm/v174/i2/p275
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