On Some Classes of Nonuniformly Elliptic Equations

  • O. A. Ladyzhenskaya
  • N. N. Ural’tseva
Part of the Seminars in Mathematics book series (SM, volume 11)

Abstract

We have called the quasi-linear equation
$${a_{ij}}(x,u,{u_x}){u_{{x_i}{x_j}}} + a(x,u,{u_x}) = 0$$
(1)
uniformly elliptic if
$$\nu (|u|){\xi ^2} \leqslant {a_{ij}}(x,u,p){\xi _i}{\xi _j} \leqslant \mu (|u|){\xi ^2}$$
where ν(τ) and µ(τ) are continuous positive functions for τ≥ 0. Such equations have been, in particu-lar, the fundamental object of investigations concerning their solvability for the case of boundary prob-lems “in the large” and the matter of obtaining various a priori estimates for all their possible solu-tions (see [2]). One of the principal a priori estimates is the estimate \(\operatorname{m} \mathop a\limits_\Omega x|{u_x}|\) . The methods we have given for obtaining this estimate (see §§3,4, Chap. 4, and §2, Chap. 6 of [2]) have also been applied, with corresponding modifications, to some classes of nonuniformly elliptic equations. A series of such classes was singled out in papers [3]–[7]. We point out here still another class of such equations.

Keywords

Elliptic Equation Consultant Bureau Fundamental Object Continuous Positive Function Global Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1970

Authors and Affiliations

  • O. A. Ladyzhenskaya
  • N. N. Ural’tseva

There are no affiliations available

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