Fritz John pp 292-300 | Cite as

# Derivatives of Solutions of Linear Elliptic Partial Differential Equations

Chapter

## 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)

*x***∂**_{1}; P is a polynomial of degree 2m in the D_{1}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(D_{1},…,D_{n}). For fixed x_{1},…,x_{n}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 (x_{1},…,_{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.## Preview

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## Bibliography

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