Uniformly Elliptic Equations in Nondivergence Form

  • Cristian E. Gutiérrez
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 89)


In this chapter we consider linear operators of the form
$$\displaystyle{Lu =\sum _{ i,j=1}^{n}a_{ ij}(x)D_{ij}u(x)}$$
where the coefficient matrix A(x) = (a ij (x)) is symmetric and uniformly elliptic, that is
$$\displaystyle{\lambda \vert \xi \vert ^{2} \leq \langle A(x)\xi,\xi \rangle \leq \Lambda \vert \xi \vert ^{2},}$$
for all \(\xi \in \mathbb{R}^{n}\) and \(x \in \Omega \subset \mathbb{R}^{n}.\)


Linear Operator Elliptic Equation Coefficient Matrix Elliptic Operator Critical Density 
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.


  1. [Cab97]
    X. Cabré, Nondivergent elliptic equations on manifolds with nonnegative curvature. Commun. Pure Appl. Math. 50 (7), 623–665 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Caf89]
    L.A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear elliptic equations. Ann. Math. 130, 189–213 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [CG97]
    L.A. Caffarelli, C.E. Gutiérrez, Properties of the solutions of the linearized Monge–Ampère equation. Am. J. Math. 119 (2), 423–465 (1997)CrossRefzbMATHGoogle Scholar
  4. [KS80]
    N.V. Krylov, M.V. Safonov, A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1), 161–175, 239 (1980)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Cristian E. Gutiérrez
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations