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Siberian Mathematical Journal

, Volume 44, Issue 4, pp 587–604 | Cite as

On the Wlq-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

  • A. P. Kopylov
Article
  • 15 Downloads

Abstract

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C l -smooth functions meet the local Hoelder condition with every exponent α, 0<α<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W l q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.

system of nonlinear partial differential equations uniformly elliptic linear system with discontinuous coefficients Wlq-regularity 

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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. P. Kopylov
    • 1
  1. 1.Sobolev Institute of MathematicsRussia

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