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Journal of Mathematical Sciences

, Volume 193, Issue 2, pp 330–339 | Cite as

On exponent in the Hölder condition for the first order derivatives of a solution to a linear elliptic second order equation with two variables

  • A. L. Treskunov
Article
  • 36 Downloads

We study local properties of weak solutions in \( W_2^2(D) \) to linear elliptic equations with measurable coefficients in the case of two independent variables. We prove that the derivatives of the weak solution to the homogeneous elliptic equation locally satisfies the Hölder condition with exponent \( {\alpha_0}=\frac{{\sqrt{33 }-3}}{2}\frac{\nu }{{{\nu^2}+1}} \), where ν ∈ (0, 1] is the ellipticity constant. For inhomogeneous equations we obtain conditions under which the derivatives of a weak solution locally satisfy the Hölder condition with exponent α < α 0. The sharpness of the results is confirmed by an example. Bibliography: 6 titles.

Keywords

Weak Solution Elliptic Equation Interior Point Order Derivative Local Property 
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References

  1. 1.
    L. Nirenberg, “On nonlinear elliptic partial differential equations and Hölder continuity,” Comm. Pure Appl. Math. 6, 103–156 (1953).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    G. Talenti, “Problemi di derivata obliqua per equazioni in due variabili” Matematiche 21, No. 2, 339–376 (1966).MathSciNetMATHGoogle Scholar
  3. 3.
    H. O. Cordes, “Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen,” Math. Ann. 131, 278–312 (1956).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    C. B. Morrey, “On the solutions of quasilinear elliptic partial differential equations,” Trans. Am. Math. Soc. 43, 126–166 (1938).MathSciNetCrossRefGoogle Scholar
  5. 5.
    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1964); English transl.: Acad. Press, New York etc. (1968).Google Scholar
  6. 6.
    A. L. Treskunov, ““On a generalization of the Weitinger inequality and its application to investigation of elliptic equations” [in Russian], Zap. Nauchn. Semin. LOMI 147, 184–187 (1985); English transl.: J. Sov. Math. 37, 905–908 (1987).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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