# Weak solution

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)