Fritz John pp 292-300 | Cite as

Derivatives of Solutions of Linear Elliptic Partial Differential Equations

Part of the Contemporary Mathematicians book series (CM)


The differential equations considered here can be written in the form
$$ L[u] = P({D_1},...,{D_n})u = f({x_1},...,{x_n}) $$
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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© Springer Science+Business Media New York 1985

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