# Weak solution

From Encyclopedia of Mathematics

*of a differential equation*

in a domain

A locally integrable function satisfying the equation

for all smooth functions (say, of class ) with compact support in . Here, the coefficients in

are assumed to be sufficiently smooth and stands for the formal Lagrange adjoint of :

For example, the generalized derivative can be defined as the locally integrable function such that is a weak solution of the equation .

In considering weak solutions of , the following problem arises: under what conditions are they strong solutions (cf. Strong solution)? For example, in the case of elliptic equations, every weak solution is strong.

#### References

[1] | A.V. Bitsadze, "Some classes of partial differential equations" , Gordon & Breach (1988) (Translated from Russian) |

#### Comments

#### References

[a1] | S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965) |

[a3] | D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1983) |

**How to Cite This Entry:**

Weak solution. A.P. Soldatov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Weak_solution&oldid=12915

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098