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The Euler equations from the point of view of differential inclusions


De Lellis, C (2008). The Euler equations from the point of view of differential inclusions. Bollettino dell'Unione Matematica Italiana, (9) 1(3):873-879.

Abstract

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity.
One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres n for n 1 and all Zoll surfaces.
Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.

Abstract

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity.
One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres n for n 1 and all Zoll surfaces.
Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:Italian
Date:2008
Deposited On:06 Feb 2009 10:20
Last Modified:24 Sep 2019 15:54
Publisher:Unione Matematica Italiana
ISSN:1972-6724
OA Status:Green
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2455350
http://arxiv.org/abs/math/0702079v3

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Content: Accepted Version
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