Fritz John pp 258-266 | Cite as

# Derivatives of Continuous Weak Solutions of Linear Elliptic Equations

Chapter

## Abstract

Let
holds;

*L*be a linear elliptic differential operator in*n*-space of order*m*. Let*M*denote the adjoint operator to*L*. An integrable function*u*is called a*weak solution*of the equation*L*[*u*] =*f*in a domain*D*, if for every*w*of class*C*_{∞}, which vanishes outside a compact subset of*D*, the relation$$
\int_D {(uM[w]) - fw)d{x_1}...d{x_n}}
$$

*u*will be called a*strict solution*, if*u*is of class*C*_{ m }and satisfies*L*[*u*] =*f*in the ordinary sense. One of the remarkable facts concerning elliptic equations is that under suitable regularity assumptions on*f*and the coefficients of*L*a weak solution can be differentiated any number of times and is a strict solution. In a recent paper F. E. Browder^{1}gives the theorem: a weak solution which is square integrable on every compact subset of*D*is almost everywhere equal to a strict solution, provided*f*is in*C*_{1}and the coefficients of the*j*-th derivatives in the operator*L*are in*C*_{ m+i }. Browder’s proof of this generalization of “Weyl’s lemma” makes use of the fundamental solution of elliptic equations with analytic coefficients.^{2}Results contained in a paper by K. O. Friedrichs,^{3}which appears in the same issue, imply the theorem that a weak solution is a strict solution, provided*f*is in*C*_{(n+1)/2}and the coefficients of*L*are in*C*_{ m/2}.## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer Science+Business Media New York 1985