# Weak solution

*of a differential equation*

$$ \tag{* } Lu \equiv \sum _ {| \alpha | \leq m } a _ \alpha ( x) D ^ \alpha u = f $$

in a domain $ D $

A locally integrable function $ u $ satisfying the equation

$$ \int\limits _ { D } u L ^ {*} \phi dx = \int\limits _ { D } f \phi dx $$

for all smooth functions $ \phi $( say, of class $ C ^ \infty $) with compact support in $ D $. Here, the coefficients $ a _ \alpha ( x) $ in

are assumed to be sufficiently smooth and $ L ^ {*} $ stands for the formal Lagrange adjoint of $ L $:

$$ L ^ {*} \phi = \ \sum _ {| \alpha | \leq m } (- 1) ^ {| \alpha | } D ^ \alpha ( a _ \alpha \phi ) . $$

For example, the generalized derivative $ f = D ^ \alpha u $ can be defined as the locally integrable function $ f $ such that $ u $ is a weak solution of the equation $ D ^ \alpha u = f $.

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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weak_solution&oldid=49184