RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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, 6Á433) 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.

Full text: PDF file (829 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1988, 60:1, 269–281

Bibliographic databases:

UDC: 517.9
MSC: 35J15, 35B45

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
\Bibitem{Saf87} \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 \mathnet{http://mi.mathnet.ru/msb1780} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=882838} \zmath{https://zbmath.org/?q=an:0656.35027} \transl \jour Math. USSR-Sb. \yr 1988 \vol 60 \issue 1 \pages 269--281 \crossref{https://doi.org/10.1070/SM1988v060n01ABEH003167} 

• http://mi.mathnet.ru/eng/msb1780
• http://mi.mathnet.ru/eng/msb/v174/i2/p275

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Dokuchaev N., “Control of Cordes-Type Diffusion with Incomplete Observation and in a Game Problem”, Differ. Equ., 32:8 (1996), 1055–1066
3. Nadirashvili N., “On the Problem of Uniqueness of Martingale”, Dokl. Akad. Nauk, 357:2 (1997), 172–175
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
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
6. R. Cavazzoni, “On the Cauchy problem for elliptic equations in a disk”, Rend Circ Mat Palermo, 52:1 (2003), 131
7. Annunziata Esposito, “An elliptic continuation result for harmonic functions in two dimensions”, Rend Circ Mat Palermo, 53:3 (2004), 437
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
9. C. G. Böhmer, “HÖLDER ESTIMATES FOR LINEAR SECOND-ORDER EQUATIONS”, Mathematika, 2010, 1
10. C. G. Böhmer, “HÖLDER ESTIMATES FOR LINEAR SECOND ORDER EQUATIONS: CORRIGENDUM”, Mathematika, 2011, 1
•  Number of views: This page: 244 Full text: 100 References: 17