of a differential equation
in a domain
A locally integrable function that has locally integrable generalized derivatives of all orders (cf. Generalized derivative), and satisfies
almost-everywhere in .
The notion of a "strong solution" can also be introduced as follows. A function is called a strong solution of
if there are sequences of smooth (for example, ) functions , such that , and for each , where the convergence is taken in for any compact set . In these definitions, can be replaced by the class of functions whose -th powers are locally integrable. The class most often used is .
In the case of an elliptic equation
both notions of a strong solution coincide.
|[a1]||J. Chazarain, A. Piriou, "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars (1981) pp. 223|
Strong solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_solution&oldid=16232