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.
|||A.V. Bitsadze, "Some classes of partial differential equations" , Gordon & Breach (1988) (Translated from Russian)|
|[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)|
Weak solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_solution&oldid=12915