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Boundedness of solutions to the Schrodinger equation under Neumann boundary conditions
University of Florence, Italy .
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2012 (English)In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, Vol. 98, no 6, 654-688 p.Article in journal (Refereed) Published
Abstract [en]

We deal with Neumann problems for Schrodinger type equations, with non-necessarily bounded potentials, in possibly irregular domains in R-n. Sharp balance conditions between the regularity of the domain and the integrability of the potential for any solution to be bounded are established. The regularity of the domain is described either through its isoperimetric function or its isocapacitary function. The integrability of the sole negative part of the potential plays a role, and is prescribed via its distribution function. The relevant conditions amount to the membership of the negative part of the potential to a Lorentz type space defined either in terms of the isoperimetric function, or of the isocapacitary function of the domain. (c) 2012 Elsevier Masson SAS. All rights reserved.

Abstract [fr]

On considère des problèmes de Neumann pour des équations de type Schrödinger, avec des potentiels non nécessairement bornés, dans des domaines éventuellement irréguliers de Rn. Des estimations optimales qui tiennent compte de la régularité du domaine et de lʼintégrabilité du potentiel sont établies afin dʼassurer que toute solution soit bornée. La régularité du domaine est décrite soit grâce à sa fonction isoperimétrique, soit grâce à sa fonction isocapacitaire. Seulement lʼintégrabilité de la partie négative du potentiel joue un rôle et elle est décrite grâce à sa fonction de distribution. La condition optimale est liée à lʼappartenance de la partie négative du potentiel à un espace de type Lorentz défini soit en termes de la fonction isopérimétrique, soit en termes de la foncion isocapacitaire du domaine.

Place, publisher, year, edition, pages
Elsevier, 2012. Vol. 98, no 6, 654-688 p.
Keyword [en]
Schrodinger equation, Neumann problem, Bounded solutions, Irregular domains, Unbounded potentials, Isocapacitary inequalities, Isoperimetric inequalities
National Category
URN: urn:nbn:se:liu:diva-87466DOI: 10.1016/j.matpur.2012.05.007ISI: 000312361900004OAI: diva2:589475

Funding Agencies|MIUR (Italian Ministry of University) via the research project "Geometric aspects of partial differential equations and related topics"||GNAMPA of INdAM (Istituto Nazionale di Alta Matematica) via the research project "Elliptic problems affected by irregularities or degenerations"||UK Engineering and Physical Sciences Research Council|EP/F005563/1|

Available from: 2013-01-18 Created: 2013-01-18 Last updated: 2013-03-13

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Maz´ya, Vladimir
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Mathematics and Applied MathematicsThe Institute of Technology
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