Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory pp 1-25 | Cite as

# Nonlocal Estimates of First Derivatives of the Solutions of the Initial Boundary Problem for Nonuniformly Elliptic and Nonuniformly Parabolic Nondivergent Equations

## Abstract

This paper is devoted to a study of nonlocal *a priori* estimates of maxima of moduli of the first derivatives of solutions of Dirichlet’s problem, and, correspondingly, the first initial-boundary problem for nonuniformly elliptic and nonuniformly parabolic nondivergent quasi-linear equations. It is closely related to known investigations of O. A. Ladyzhenskaya and N. N. Ural’tseva on quasi-linear elliptic and parabolic equations and systems [1, 2]. A characteristic peculiarity of the paper is the fact that the method, developed by O. A. Ladyzhenskaya and N. N. Ural’tseva, for obtaining *a priori* estimates of maxima of moduli of the first derivatives for solutions of uniformly elliptic and uniformly parabolic quasi-linear equations with divergent principal part, is used here for studying analogous estimates for solutions of nondivergent equations; moreover, the method enables one to investigate specific classes of nonuniformly elliptic and nonuniformly parabolic quasi-linear equations, including those not belonging to S. N. Bernshtein’s class (L).

## Keywords

Volume Integral Surface Integral Cauchy Inequality Parabolic Case Obvious Inequality## Preview

Unable to display preview. Download preview PDF.

## Literature Cited

- 1.O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasi-linear Equations of Elliptic Type, Mathematics in Science and Engineering, Vol. 46, Academic Press, New York (1968).Google Scholar
- 2.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasi-linear Equations of Parabolic Type, “Nauka,” Moscow (1967).Google Scholar
- 3.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. 12–34.Google Scholar
- 4.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
- 5.N. M. Ivochkina and A. P. Oskolkov, “On estimates of the first derivatives for nonuniformly elliptic nondivergent equations,” Problems of Applied Mathematics and Geometrical Modeling, Leningrad Institute of Engineering and Construction, Leningrad (1968).Google Scholar
- 6.N. M. Ivochkina, “The Dirichlet problem for two-dimensional quasi-linear second-order elliptic equations,” Problems of Mathematical Analysis, Vol. 2, Leningrad State University, Leningrad (1968) [English translation: Consultants Bureau, New York (1971)].Google Scholar
- 7.N. Meyers, “On a class of nonuniformly elliptic quasi-linear equations in the plane,” Arch. Rat. Mech. Anal., Vol. 12, No. 5, pp. 367–391 (1963).CrossRefGoogle Scholar
- 8.T. B. Solomyak, “Dirichlet problem for nonuniformly elliptic quasi-linear equations,” Problems of Mathematical Analysis, Vol. 1, Boundary Value Problems and Integral Equations, Consultants Bureau, New York (1968), pp. 81–101.Google Scholar
- 9.N. M. Ivochkina, The First Boundary Problem for Nonuniformly Elliptic and Nonuniformly Parabolic Quasi-linear Equations, Candidate’s Dissertation, Leningrad Branch of the Academy of Sciences of the USSR (1968).Google Scholar
- 10.N. S. Trudinger, “Quasi-linear elliptic partial differential equations in n variables,” Department of Mathematics, Stanford University, Stanford (July, 1966 ).Google Scholar
- 11.J. Serrin, “The Dirichlet problem for quasi-linear equations with many independent variables,” Proc. Nat. Acad. Sci. USA, Vol. 58, No. 5 (1967).Google Scholar