# Lavrent'ev theorem

Lavrent'ev's theorem in descriptive set theory: A topological mapping between two sets in $\mathbf R ^ {n}$ can be extended to a homeomorphism between certain sets of type $G _ \delta$ containing them. A consequence of this theorem is that the Hausdorff type of a set is a topological invariant (see ).

Lavrent'ev's theorem in approximation theory gives a criterion for the possibility of uniform approximation: In order that a function, continuous on a compact set $K \subset \mathbf C$, can be uniformly approximated on $K$ by polynomials it is necessary and sufficient that $K$ is a compact set without interior points that does not partition the complex plane (see ).

Lavrent'ev's theorem in the theory of quasi-conformal mapping: Let $D _ {z}$ and $D _ {w}$ be two simply-connected domains in the plane bounded by piecewise-smooth curves and let $z _ {1} , z _ {2} , z _ {3}$ and $w _ {1} , w _ {2} , w _ {3}$ be triples of positively enumerated points on their boundaries. Then for any strongly-elliptic system of equations

$$\Phi _ {1} ( x , y , u , v , u _ {x} ^ \prime ,\ u _ {y} ^ \prime , v _ {x} ^ \prime , v _ {y} ^ \prime ) = 0 ,$$

$$\Phi _ {2} ( x , y , u , v , u _ {x} ^ \prime , u _ {y} ^ \prime , v _ {x} ^ \prime , v _ {y} ^ \prime ) = 0 ,$$

with uniformly continuous partial derivatives of the functions that specify the equations of the characteristics, there is always a unique homeomorphic mapping of $D _ {z}$ onto $D _ {w}$ that realizes a solution $u ( x , y )$, $v ( x , y )$ of the system, and under which the given triples of boundary points correspond to each other.

For Lavrent'ev's theorem in mechanics (aerofoil theory, solitary wave, forms of dynamical loss of stability, flows, the theory of cumulative charge, directed detonation) see .

Theorems 1)–4) are due to M.A. Lavrent'ev.

How to Cite This Entry:
Lavrent'ev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lavrent%27ev_theorem&oldid=47593
This article was adapted from an original article by V.A. Zorich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article