Timeline for Are there alternative proofs for existence/uniqueness of ODE solutions?
Current License: CC BY-SA 4.0
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| Jul 22, 2022 at 6:07 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
minor typos
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| Aug 27, 2010 at 9:36 | comment | added | Johannes Hahn |
No, not necessarily. As I said a solution to some PDE can often be obtained as a (the) minimum of a functional {Space of functions}$\to\mathbb{R}$. For example $u \mapsto \int_\Omega \|\nabla u\|_2^2$ has a unique minimum on $\lbrace u\in W^{1,2}(\Omega) | u_{|\partial\Omega}=g\rbrace$ if $g$ satisfies a certain condition. This minimum is exactly the solution of the Dirichlet equation $\Delta u=0, u_{|\partial\Omega}=g$. Existence and uniqueness of this minimum can be proved without any local considerations.
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| Aug 27, 2010 at 3:11 | comment | added | Sujit_Nair | I was considering variational principle as an "alternative". But the, the local existence/uniqueness for solutions of variational principle again relies on local existence/uniqueness of ODEs in a coordinate chart. Right? | |
| Aug 26, 2010 at 18:56 | history | edited | Johannes Hahn | CC BY-SA 2.5 |
added 955 characters in body
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| Aug 26, 2010 at 18:49 | history | answered | Johannes Hahn | CC BY-SA 2.5 |