# Weak solution

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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)

#### 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
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article