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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 2, Pages 275–288 (Mi msb1780)  

This article is cited in 10 scientific papers (total in 10 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 $l=(l_1,…,l_n)\in E_n$ ($n\geqslant2$, $\nu=\mathrm{const}>0$). 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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English version:
Mathematics of the USSR-Sbornik, 1988, 60:1, 269–281

Bibliographic databases:

UDC: 517.9
MSC: 35J15, 35B45
Received: 17.09.1985

Citation: M. V. Safonov, “Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients”, Mat. Sb. (N.S.), 132(174):2 (1987), 275–288; Math. USSR-Sb., 60:1 (1988), 269–281

Citation in format AMSBIB
\by M.~V.~Safonov
\paper Unimprovability of estimates of H\"older constants for solutions of linear elliptic equations with measurable coefficients
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 2
\pages 275--288
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 1
\pages 269--281

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    This publication is cited in the following articles:
    1. Bass R., “The Dirichlet Problem for Radially Homogeneous Elliptic-Operators”, Trans. Am. Math. Soc., 320:2 (1990), 593–614  crossref  mathscinet  zmath  isi
    2. Dokuchaev N., “Control of Cordes-Type Diffusion with Incomplete Observation and in a Game Problem”, Differ. Equ., 32:8 (1996), 1055–1066  mathnet  mathscinet  zmath  isi
    3. Nadirashvili N., “On the Problem of Uniqueness of Martingale”, Dokl. Akad. Nauk, 357:2 (1997), 172–175  mathnet  mathscinet  zmath  isi
    4. P. Buonocore, P. Manselli, “Solutions to two dimensional, uniformly elliptic equations, that lie in sobolev spaces and have compact support”, Rend Circ Mat Palermo, 51:3 (2002), 476  crossref  mathscinet  zmath
    5. Qing Han, Nikolai Nadirashvili, Yu Yuan, “Linearity of homogeneous order-one solutions to elliptic equations in dimension three”, Comm Pure Appl Math, 56:4 (2003), 425  crossref  mathscinet  zmath
    6. R. Cavazzoni, “On the Cauchy problem for elliptic equations in a disk”, Rend Circ Mat Palermo, 52:1 (2003), 131  crossref  mathscinet  zmath
    7. Annunziata Esposito, “An elliptic continuation result for harmonic functions in two dimensions”, Rend Circ Mat Palermo, 53:3 (2004), 437  crossref  mathscinet  zmath
    8. Cristina Giannotti, “A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term”, Journal of Differential Equations, 201:2 (2004), 234  crossref  mathscinet  zmath
    9. C. G. Böhmer, “HÖLDER ESTIMATES FOR LINEAR SECOND-ORDER EQUATIONS”, Mathematika, 2010, 1  crossref  mathscinet
    10. C. G. Böhmer, “HÖLDER ESTIMATES FOR LINEAR SECOND ORDER EQUATIONS: CORRIGENDUM”, Mathematika, 2011, 1  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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