Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory pp 47-53 | Cite as

# On Some Classes of Nonuniformly Elliptic Equations

Chapter

## Abstract

We have called the quasi-linear equation
uniformly elliptic if
where

$${a_{ij}}(x,u,{u_x}){u_{{x_i}{x_j}}} + a(x,u,{u_x}) = 0$$

(1)

$$\nu (|u|){\xi ^2} \leqslant {a_{ij}}(x,u,p){\xi _i}{\xi _j} \leqslant \mu (|u|){\xi ^2}$$

*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
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## Literature Cited

- 1.O. A. Ladyzhenskaya and N. N. Ural’tseva, “Certain classes of nonuniformly elliptic equations,” Seminars in Mathematics, Vol. 5, Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory, Part I, Consultants Bureau, New York (1969), pp. 67–69.Google Scholar
- 2.O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasi-linear Elliptic Equations, Mathematics in Science and Engineering, Vol. 46, Academic Press, New York (1968).Google Scholar
- 3.A. P. Oskolkov, “On some estimates for nonuniformly elliptic equations and systems,” Trudy Matem. inst. im V. A. Steklova, Vol. 92, pp. 203–232 (1966).Google Scholar
- 4.A. P. Oskolkov,
*A priori*estimates of the first derivatives of solutions of Dirichlet’s problem for nonuniformly elliptic quasi-linear equations,“ TrudyMatem. inst. im. V. A. Steklova, Vol. 102 (1967).Google Scholar - 5.N. M. Ivochkina, “Dirichlet’s problem for two-dimensional quasi-linear second-order elliptic equations,” in Problems of Mathematical Analysis, Collected Papers [in Russian], No. 2, Leningrad State Univ., Leningrad (1968) [English translation: Consultants Bureau, New York (in preparation).]Google Scholar
- 6.N. M. Ivochkina and A. P. Oskolkov, “Nonlocal estimates of first derivatives of solutions of Dirichlet’s problem for nonuniformly elliptic quasi-linear equations,” Seminars in Mathematics, Vol. 5, Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory, Part I, Consultants Bureau, New York (1969), pp. 12–35.Google Scholar
- 7.N. M. Ivochkina and A. P. Oskolkov, “Nonlocal estimates of first derivatives of the solutions of the initial-boundary problem for nonuniformly elliptic and nonuniformly parabolic nondivergent equations,” this volume, p. 1.Google Scholar
- 8.I. Ya. Bakel’man, “The construction of surfaces with given mean curvature and associated problems of the theory of quasi-linear equations,” Reports at the Second All-Union Symposium on Global Geometry [in Russian], Petrozavodsk (1967), pp. 9–10.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1970