Difference between revisions of "Strong solution"
From Encyclopedia of Mathematics
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| − | ''of a differential equation | + | {{TEX|done}} |
| + | ''of a differential equation'' | ||
| − | + | $$Lu\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha u=f\label{*}\tag{*}$$ | |
| − | in a domain | + | ''in a domain $D$'' |
| − | A locally integrable function | + | A locally integrable function $u$ that has locally integrable generalized derivatives of all orders $\leq m$ (cf. [[Generalized derivative|Generalized derivative]]), and satisfies \eqref{*} almost-everywhere in $D$. |
| − | + | The notion of a "strong solution" can also be introduced as follows. A function $u$ is called a strong solution of \eqref{*} if there are sequences of smooth (for example, $C^\infty$) functions $\{u_n\}$, $\{f_n\}$ such that $u_n\to u$, $f_n\to f$ and $Lu_n=f_n$ for each $n$, where the convergence is taken in $L_1(K)$ for any compact set $K\subseteq D$. In these definitions, $L_1$ can be replaced by the class $L_p$ of functions whose $p$-th powers are locally integrable. The class most often used is $L_2$. | |
| − | + | In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide. | |
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| − | In the case of an elliptic equation | ||
| − | |||
| − | both notions of a strong solution coincide. | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Chazarain, A. Piriou, "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars (1981) pp. 223</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Chazarain, A. Piriou, "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars (1981) pp. 223 {{MR|0598467}} {{ZBL|0446.35001}} </TD></TR></table> |
Latest revision as of 15:58, 14 February 2020
of a differential equation
$$Lu\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha u=f\label{*}\tag{*}$$
in a domain $D$
A locally integrable function $u$ that has locally integrable generalized derivatives of all orders $\leq m$ (cf. Generalized derivative), and satisfies \eqref{*} almost-everywhere in $D$.
The notion of a "strong solution" can also be introduced as follows. A function $u$ is called a strong solution of \eqref{*} if there are sequences of smooth (for example, $C^\infty$) functions $\{u_n\}$, $\{f_n\}$ such that $u_n\to u$, $f_n\to f$ and $Lu_n=f_n$ for each $n$, where the convergence is taken in $L_1(K)$ for any compact set $K\subseteq D$. In these definitions, $L_1$ can be replaced by the class $L_p$ of functions whose $p$-th powers are locally integrable. The class most often used is $L_2$.
In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide.
Comments
References
| [a1] | J. Chazarain, A. Piriou, "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars (1981) pp. 223 MR0598467 Zbl 0446.35001 |
How to Cite This Entry:
Strong solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_solution&oldid=16232
Strong solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_solution&oldid=16232
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article