Another way to say subsolution: the maximum principle and sums of Green functions
Keyword(s):
Consider an elliptic second order differential operator $L$ with no zeroth order term (for example the Laplacian $L=-\Delta$). If $Lu \leq 0$ in a domain $U$, then of course $u$ satisfies the maximum principle on every subdomain $V \subset U$. We prove a converse, namely that $Lu \leq 0$ on $U$ if on every subdomain $V$, the maximum principle is satisfied by $u+v$ whenever $v$ is a finite linear combination (with positive coefficients) of Green functions with poles outside $\overline{V}$. This extends a result of Crandall and Zhang for the Laplacian. We also treat the heat equation, improving Crandall and Wang's recent result. The general parabolic case remains open.
1986 ◽
Vol 108
(4)
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pp. 330-339
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2017 ◽
Vol 55
(5)
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pp. 2905-2935
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Keyword(s):
2008 ◽
Vol 18
(04)
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pp. 511-541
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1990 ◽
Vol 65
(1)
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pp. 255-270
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