Elliptic Operators

  • Lester L. HelmsEmail author
Part of the Universitext book series (UTX)


In previous chapters the emphasis was on the Dirichlet and Neumann Problems associated with the Laplacian for which mixed boundary conditions were allowed. In this chapter the Laplacian is replaced by an elliptic operator of the form
$$\mathbf {L} u(x) = \sum _{i,j=1}^{n}a_{ij}(x)\frac{\partial ^2 u}{\partial x_i \partial x_j}(x) + \sum _{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}(x) + c(x)u(x), x \in \Omega $$
and boundary conditions are replaced by an first order differential operator
$$\mathbf {M}u(x) = \sum _{i=1}^{n}\beta _i(x)\frac{\partial u}{\partial x_i}(x) + \gamma (x)u(x) = 0, x \in \partial \Omega .$$
The Dirichlet and Neumann are is special cases of the problem
$$ \mathbf {L} = f \text{ on } \Omega \subset R^n $$
subject to the boundary conditions \(u = g \text{ on } \partial \Omega \) and \(\mathbf {M} = g\) on \(\partial \Omega \), respectively. The latter is known as the oblique derivative boundary problem. To transfer classical results to a more general setting, a method of continuity is used. Proceeding from an elliptic operator with constant coefficients, classical Dirichlet solutions are morphed into solutions on a ball for the operator \(\mathbf {L}\). The Perron-Brelot-Wiener method is then adapted to extend results to the more general setting.


Elliptic Operator Mixed Boundary Conditions Exterior Sphere Condition Weak Maximum Principle Superfunction 
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  1. 1.
    Hohn, F.E.: Elementary Matrix Algebra. The Macmillan Company, New York (1973)Google Scholar
  2. 2.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1983)CrossRefzbMATHGoogle Scholar
  3. 3.
    Apostol, T.M.: Mathematical Analysis. Addison-Wesley Publishing Company, Reading (1957)zbMATHGoogle Scholar
  4. 4.
    Hopf, E.: A remark on linear elliptic differential equations of second order. Proc. Am. Math. Soc. 3, 791–793 (1952)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hopf, E.: Elementare bemerkungen über die lösungen partieller differentialgleichungen zweiter ordnung vom elliptischen typus. Sitz. Ber. Preuss Akad. Wissensch. Berlin Math. Phys. 19, 147–152 (1927)Google Scholar
  6. 6.
    Hervé, R.M.: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel. Ann. Inst. Fourier (Grenoble) 12, 415–571 (1962)Google Scholar
  7. 7.
    Poincaré, H.: Sur les equations aux dérivées partielles de la physique mathématique. Am. J. Math. 12, 211–294 (1890)Google Scholar
  8. 8.
    Miller, K.: Barriers for cones for uniformly elliptic operators. Ann. Mat. Pura. Appl. (Series 4), 76, 93–105 (1967)Google Scholar

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© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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