Borel's lemma

In mathematics, Borel's lemma is an important result about partial differential equations named after Émile Borel.

Suppose $U$ is an open set in the Euclidean space Rⁿ, and suppose that $f_{0},f_{1},...$ is a sequence of smooth, complex-valued functions on $U$. Then there exists a smooth function $F=F(t,x)$ defined on R×$U$ with complex values, such that

$\left({\frac {\partial ^{k}}{\partial t^{k}}}F\right)(0,x)=f_{k}(x),$

for all $k=0,1,...$, and $x$ in $U.$

A constructive proof of this result is given in Golubitsky (1974).

References

• M. Golubitsky, V. Guillemin (1974). Stable mappings and their singularities. Springer-Verlag, Graduate texts in Mathematics: Vol. 14. ISBN 0-387-90072-1.

Borel lemma at PlanetMath.