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Selected Topics in Partial Differential Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This Ph.D. thesis consists of five papers and an introduction to the main topics of the thesis. In Paper I we give an abstract criteria for existence of multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators. In paper II we establish existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations for Coulomb systems along with properties of the solutions. In Paper III we establish existence of a ground state to the magnetic Hartree-Fock equations. In Paper IV we study the Choquard equation with general potentials (including quasirelativistic and magnetic versions of the equation) and establish existence of multiple solutions. In Paper V we prove that, under some assumptions on its nonmagnetic counterpart, a magnetic Schrödinger operator admits a representation with a positive Lagrange density and we derive consequences of this property.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2011. , x, 14 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 70
National Category
Mathematics Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-145763ISBN: 978-91-506-2193-8OAI: oai:DiVA.org:uu-145763DiVA: diva2:398102
Public defence
2011-03-31, Häggsalen, Lägerhyddsvägen 1, Uppsala, 09:15 (English)
Opponent
Supervisors
Note
I den tryckta boken har förlag felaktigt angivits som Acta Universitatis Upsaliensis.Available from: 2011-03-10 Created: 2011-02-10 Last updated: 2011-10-25Bibliographically approved
List of papers
1. Abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators
Open this publication in new window or tab >>Abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators
2012 (English)In: Electronic Journal of Differential Equations, ISSN 1072-6691, Vol. 253, no 6, 1729-1743 p.Article in journal (Refereed) Published
Abstract [en]

We consider a system of nonlinear coupled equations involving magnetic Schrodinger operators and general potentials. We provide the criteria for the existence of multiple solutions to these equations. As special cases we get the classical results on existence of infinitely many distinct solutions within Hartree and Hartree-Fock theory of atoms and molecules subject to an external magnetic fields. We also extend recent results within this theory, including Coulomb system with a constant magnetic field, a decreasing magnetic field and a "physically measurable" magnetic field.

Keyword
Nonlinear equations, Hartree-Fock model, Magnetic fields, Excited states, Ground state, Critical point theory
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-145759 (URN)10.1016/j.jde.2012.05.025 (DOI)000306718400004 ()
Available from: 2011-02-10 Created: 2011-02-10 Last updated: 2012-08-28Bibliographically approved
2. Existence of infinitely many solutions to the quasi-relativistic Hartree-Fock equations
Open this publication in new window or tab >>Existence of infinitely many solutions to the quasi-relativistic Hartree-Fock equations
2011 (English)Report (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-145761 (URN)
Available from: 2011-02-10 Created: 2011-02-10 Last updated: 2012-02-16Bibliographically approved
3. Existence of a solution to Hartree-Fock equations with decreasing magnetic fields
Open this publication in new window or tab >>Existence of a solution to Hartree-Fock equations with decreasing magnetic fields
2008 (English)Report (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-145762 (URN)
Available from: 2011-02-10 Created: 2011-02-10 Last updated: 2012-05-16Bibliographically approved
4. Multiple solutions of Choquard type equations
Open this publication in new window or tab >>Multiple solutions of Choquard type equations
2010 (English)Report (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-145760 (URN)
Available from: 2011-02-10 Created: 2011-02-10 Last updated: 2012-05-16Bibliographically approved
5. Weighted spectral gap for magnetic Schrödinger operators with a potential term
Open this publication in new window or tab >>Weighted spectral gap for magnetic Schrödinger operators with a potential term
2009 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 31, no 3, 215-226 p.Article in journal (Refereed) Published
Abstract [en]

We prove that singular Schrodinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their "nonmagnetic" counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L (p) -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrodinger equation with the periodic magnetic field.

Keyword
Schrodinger equation, External magnetic field, Positive solutions, Spectral gap, Ground state
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-120897 (URN)10.1007/s11118-009-9132-x (DOI)000269915200002 ()
Available from: 2010-03-17 Created: 2010-03-17 Last updated: 2011-10-25Bibliographically approved

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