Integro-differential equations : regularity theory and Pohozaev identities
The main topic of the thesis is the study of Elliptic PDEs. It is divided into three parts: (I) integro-differential equations, (II) stable solutions to reaction-diffusion problems, and (III) weighted isoperimetric and Sobolev inequalities. Integro-differential equations arise naturally in the study...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/279289 |
| Acceso en línea: | http://hdl.handle.net/10803/279289 https://dx.doi.org/10.5821/dissertation-2117-95453 |
| Access Level: | acceso abierto |
| Palabra clave: | 51 |
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Integro-differential equations : regularity theory and Pohozaev identities |
| title |
Integro-differential equations : regularity theory and Pohozaev identities |
| spellingShingle |
Integro-differential equations : regularity theory and Pohozaev identities Ros, Xavier 51 |
| title_short |
Integro-differential equations : regularity theory and Pohozaev identities |
| title_full |
Integro-differential equations : regularity theory and Pohozaev identities |
| title_fullStr |
Integro-differential equations : regularity theory and Pohozaev identities |
| title_full_unstemmed |
Integro-differential equations : regularity theory and Pohozaev identities |
| title_sort |
Integro-differential equations : regularity theory and Pohozaev identities |
| dc.creator.none.fl_str_mv |
Ros, Xavier |
| author |
Ros, Xavier |
| author_facet |
Ros, Xavier |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Cabré, Xavier Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I |
| dc.subject.none.fl_str_mv |
51 |
| topic |
51 |
| description |
The main topic of the thesis is the study of Elliptic PDEs. It is divided into three parts: (I) integro-differential equations, (II) stable solutions to reaction-diffusion problems, and (III) weighted isoperimetric and Sobolev inequalities. Integro-differential equations arise naturally in the study of stochastic processes with jumps, and are used in Finance, Physics, or Ecology. The most canonical example of integro-differential operator is the fractional Laplacian (the infinitesimal generator of the radially symmetric stable process). In the first Part of the thesis we find and prove the Pohozaev identity for such operator. We also obtain boundary regularity results for general integro-differential operators, as explained next. In the classical case of the Laplacian, the Pohozaev identity applies to any solution of linear or semilinear problems in bounded domains, and is a very important tool in the study of elliptic PDEs. Before our work, a Pohozaev identity for the fractional Laplacian was not known. It was not even known which form should it have, if any. In this thesis we find and establish such identity. Quite surprisingly, it involves a local boundary term, even though the operator is nonlocal. The proof of the identity requires fine boundary regularity properties of solutions, that we also establish here. Our boundary regularity results apply to fully nonlinear integro-differential equations, but they improve the best known ones even for linear ones. Our work in Part II concerns the regularity of local minimizers to some elliptic equations, a classical problem in the Calculus of Variations. More precisely, we study the regularity of stable solutions to reaction-diffusion problems in bounded domains. It is a long standing open problem to prove that all stable solutions are bounded, and thus regular, in dimensions n<10. In dimensions n>=10 there are examples of singular stable solutions. The question is still open in dimensions 4<n<10. We prove here that, in domains of double revolution, all stable solutions are bounded in dimensions n<8. Except for the radial case, our result is the first partial answer valid for all nonlinearities. While studying this, we were led to some weighted Sobolev inequalities with monomial weights that were not treated in the literature. We establish them in Part III of the thesis. Our proof of such Sobolev inequalities is based on a new weighted isoperimetric inequality in R^n. It is quite surprising that, even if the weight is not radially symmetric, Euclidean balls (centered at the origin) solve this isoperimetric problem. Also in Part III, we study more general weights, and also anisotropic perimeters. We obtain a family of new isoperimetric inequalities with homogeneous weights satisfying a concavity condition. As a particular case of our results, we provide with totally new proofs of two classical results: the Wulff inequality, and the Lions-Pacella inequality. |
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2014 |
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2014 2014 2014 |
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info:eu-repo/semantics/doctoralThesis info:eu-repo/semantics/publishedVersion |
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doctoralThesis |
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publishedVersion |
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http://hdl.handle.net/10803/279289 https://dx.doi.org/10.5821/dissertation-2117-95453 |
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http://hdl.handle.net/10803/279289 https://dx.doi.org/10.5821/dissertation-2117-95453 |
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Inglés |
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Inglés |
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http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess |
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http://creativecommons.org/licenses/by/3.0/es/ |
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openAccess |
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309 p. application/pdf application/pdf |
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Universitat Politècnica de Catalunya |
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Universitat Politècnica de Catalunya |
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TDX (Tesis Doctorals en Xarxa) reponame:TDR. Tesis Doctorales en Red instname:CBUC, CESCA |
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CBUC, CESCA |
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TDR. Tesis Doctorales en Red |
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TDR. Tesis Doctorales en Red |
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1869422958694367232 |
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Integro-differential equations : regularity theory and Pohozaev identitiesRos, Xavier51The main topic of the thesis is the study of Elliptic PDEs. It is divided into three parts: (I) integro-differential equations, (II) stable solutions to reaction-diffusion problems, and (III) weighted isoperimetric and Sobolev inequalities. Integro-differential equations arise naturally in the study of stochastic processes with jumps, and are used in Finance, Physics, or Ecology. The most canonical example of integro-differential operator is the fractional Laplacian (the infinitesimal generator of the radially symmetric stable process). In the first Part of the thesis we find and prove the Pohozaev identity for such operator. We also obtain boundary regularity results for general integro-differential operators, as explained next. In the classical case of the Laplacian, the Pohozaev identity applies to any solution of linear or semilinear problems in bounded domains, and is a very important tool in the study of elliptic PDEs. Before our work, a Pohozaev identity for the fractional Laplacian was not known. It was not even known which form should it have, if any. In this thesis we find and establish such identity. Quite surprisingly, it involves a local boundary term, even though the operator is nonlocal. The proof of the identity requires fine boundary regularity properties of solutions, that we also establish here. Our boundary regularity results apply to fully nonlinear integro-differential equations, but they improve the best known ones even for linear ones. Our work in Part II concerns the regularity of local minimizers to some elliptic equations, a classical problem in the Calculus of Variations. More precisely, we study the regularity of stable solutions to reaction-diffusion problems in bounded domains. It is a long standing open problem to prove that all stable solutions are bounded, and thus regular, in dimensions n<10. In dimensions n>=10 there are examples of singular stable solutions. The question is still open in dimensions 4<n<10. We prove here that, in domains of double revolution, all stable solutions are bounded in dimensions n<8. Except for the radial case, our result is the first partial answer valid for all nonlinearities. While studying this, we were led to some weighted Sobolev inequalities with monomial weights that were not treated in the literature. We establish them in Part III of the thesis. Our proof of such Sobolev inequalities is based on a new weighted isoperimetric inequality in R^n. It is quite surprising that, even if the weight is not radially symmetric, Euclidean balls (centered at the origin) solve this isoperimetric problem. Also in Part III, we study more general weights, and also anisotropic perimeters. We obtain a family of new isoperimetric inequalities with homogeneous weights satisfying a concavity condition. As a particular case of our results, we provide with totally new proofs of two classical results: the Wulff inequality, and the Lions-Pacella inequality.El tema principal de la tesi és l'estudi d'EDPs el·líptiques. La tesi està dividida en tres parts: (I) equacions integro-diferencials, (II) solucions estables de problemes de reacció-difusió, i (III) desigualtats isoperimètriques i de Sobolev amb pesos. Les equacions integro-differencials apareixen de manera natural en l'estudi de processos estocàstics amb salts (processos de Lévy), i s'utilitzen per modelitzar problemes en Finances, Física, o Ecologia. L'exemple més canònic d'operador integro-diferencial és el Laplacià fraccionari (el generador infinitesimal d'un procés estable i radialment simètric). A la Part I de la tesi trobem i demostrem la identitat de Pohozaev per aquest operador. També obtenim resultats de regularitat a la vora per operadors integro-diferencials més generals, tal com expliquem a continuació. En el cas clàssic del Laplacià, la identitat de Pohozaev s'aplica a qualsevol solució de problemes lineals o semilineals en dominis acotats, i és una eina molt important en l'estudi d'EDPs el·líptiques. Abans del nostre treball, no es coneixia cap identitat de Pohozaev pel Laplacià fraccionari. Ni tan sols es sabia quina forma hauria de tenir, en cas que existís. En aquesta tesi trobem i demostrem aquesta identitat. Sorprenentment, la identitat involucra un terma de vora local, tot i que l'operador és no-local. La demostració de la identitat requereix conèixer el comportament precís de les solucions a la vora, cosa que també obtenim aquí. Els nostres resultats de regularitat a la vora s'apliquen a equacions integro-diferencials completament no-lineals, però milloren els resultats anteriors fins i tot per a equacions lineals. A la Part II estudiem la regularitat dels minimitzants locals d'algunes equacions el·líptiques, un problema clàssic del Càlcul de Variacions. En concret, estudiem la regularitat de les solucions estables a problemes de reacció-difusió en dominis acotats. És un problema obert des de fa molts anys demostrar que totes les solucions estables són acotades (i per tant regulars) en dimensions n<10. En dimensions n>=10 hi ha exemples de solucions estables singulars. La questió encara està oberta en dimensions 4<n<10. Aquí demostrem que, en dominis de doble revolució, totes les solucions estables són acotades en dimensions n<8. Excepte el cas radial, el nostre resultat és la primera resposta parcial vàlida per a totes les nolinealitats en dimensions 5, 6 i 7. Mentre estudiavem aquest problema, ens vam trobar amb desigualtats de Sobolev amb pesos monomials que no havien estat tractades a la literatura. A la Part III, demostrem aquestes desigualtats. La nostra demostració d'aquestes desigualtats de Sobolev es basa en una nova desigualtat isoperimètrica a R^n amb un pes. És bastant sorprenent que, tot i que el pes no és radialment simètric, les boles (centrades a l'origen) resolen aquest problema isoperimètric. També a la Part III, estudiem pesos més generals, i també perímetres no-isotròpics. Obtenim una nova família de desigualtats isoperimètriques amb pesos homogenis que satisfan una condició de concavitat. Com a cas particular dels nostres resultats, donem demostracions totalment noves de dos resultats clàssics: la desigualtat de Wulff, i la desigualtat de Lions-Pacella.DOCTORAT EN MATEMÀTICA APLICADA (Pla 2007)Universitat Politècnica de CatalunyaCabré, XavierUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I201420142014info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersion309 p.application/pdfapplication/pdfhttp://hdl.handle.net/10803/279289https://dx.doi.org/10.5821/dissertation-2117-95453TDX (Tesis Doctorals en Xarxa)reponame:TDR. Tesis Doctorales en Redinstname:CBUC, CESCAInglésL'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by/3.0/es/http://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccessoai:www.tdx.cat:10803/2792892026-06-14T12:46:07Z |
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15,300719 |