Weak solution
From Encyclopedia of Mathematics
of a differential equation
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in a domain
A locally integrable function satisfying the equation
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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
:
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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.
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
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