On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations 论文

摘要

In this paper, the analogues of the theorems of [4], [5 1 oIn solutions of elliptic partial differential equations will be obtained for the case where the lnumber of independent variables exceeds 2. For the sake of notational simplicity, it will be assumed that the number of independent variables is 3. The equation to be considered is of the type Au + 0, where Au is the Euclidean Laplacian of i, and no second order partial derivative of i occurs in the rest of the equation. The replacement of Au by a more general linear combinationi of second derivatives of u will not be considered at this time. (In the plane, this more general case can be reduced to the special case by colnformal mappings under suitable smoothness assumptions on the coefficients; in space, perturbatioin imiethods of Korn and Lichtenstein can be used.) For simplicitv, the partial differential equation to be considered will be assumed to be linear (the methods are applicable to non-linear equations of the type (5)-(6) below). The first part of the paper deals with solutions near a zero, the second part with solutions near an isolated singularity.