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Fritz John pp 292-300 | Cite as

Derivatives of Solutions of Linear Elliptic Partial Differential Equations

Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

The differential equations considered here can be written in the form
$$ L[u] = P({D_1},...,{D_n})u = f({x_1},...,{x_n}) $$
(1)
x1; P is a polynomial of degree 2m in the D1 with coefficients which are functions of the independent variables and f is a given function. The polynomial P can be thought of as a sum of homogeneous polynomials with degrees varying from 2m down to 0. Here the polynomial consisting of the terms of the highest degree 2m in P is to be called the “principal part” of P, and will be denoted by Q(D1,…,Dn). For fixed x1,…,xn the expression Q(ξ 1 ,…,ξ n ) considered as a function of the variables ξ 1 is the characteristic form of the differential equation at the point (x1,…,xn) = x. The equation is elliptic, if the form Q is definite for all x in question. In this case the order of the equation is necessarily even, and hence m is an integer.

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Bibliography

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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • F. John

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