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  • Propagation in reaction-diffusion equations with fractional diffusion  Open access

     Coulon, Anne-charline
    Universitat Politècnica de Catalunya
    Theses

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    Esta tesis se centra en el comportamiento en tiempos grandes de las soluciones de la ecuación de Fisher- KPP de reacción-difusión con difusión fraccionaria. Este tipo de ecuación surge, por ejemplo, en la propagación espacial o en la propagación de especies biológicas (ratas, insectos,...). En la dinámica de poblaciones, la cantidad que se estudia representa la densidad de la población.Es conocido que, bajo algunas hipótesis específicas, la solución tiende a un estado estable del problema de evolución, cuando el tiempo tiende a infinito. En otras palabras, la población invade el medio, lo que corresponde a la supervivencia de la especie, y nosotros queremos entender con qué velocidad se lleva a cabo esta invasión. Para responder a esta pregunta, hemos creado un nuevo método para estudiar la velocidad de propagación cuando se consideran difusiones fraccionarias, además hemos aplicado este método en tres problemas diferentes. La Parte I de la tesis está dedicada al análisis de la ubicación asintótica de los conjuntos de nivel de la solución de dos problemas diferentes: modelos de Fisher- KPP en medios periódicos y sistemas cooperativos, ambos consideran difusión fraccionaria. En el primer modelo, se prueba que, bajo ciertas hipótesis sobre el medio periódico, la solución se propaga exponencialmente rápido en el tiempo, además encontramos el exponente exacto que aparece en esta velocidad de propagación exponencial. También llevamos a cabo simulaciones numéricas para investigar la dependencia de la velocidad de propagación con la condición inicial. En el segundo modelo, se prueba que la velocidad de propagación es nuevamente exponencial en el tiempo, con un exponente que depende del índice más pequeño de los Laplacianos fraccionarios y también del término de reacción.La Parte II de la tesis ocurre en un entorno de dos dimensiones, donde se reproduce un tipo ecuación de Fisher- KPP con difusión estándar, excepto en una línea del plano, en el que la difusión fraccionada aparece. El plano será llamado "campo" y la línea "camino", como una referencia a las situaciones biológicas que tenemos en mente. De hecho, desde hace tiempo se sabe que la difusión rápida en los caminos puede causar un efecto en la propagación de epidemias. Probamos que la velocidad de propagación es exponencial en el tiempo en el camino, mientras que depende linealmente del tiempo en el campo. Contrariamente a los precisos exponentes obtenidos en la Parte I, para este modelo, no fuimos capaces de dar una localización exacta de los conjuntos de nivel en la carretera y en el campo. La forma de propagación de los conjuntos de nivel en el campo se investiga a través de simulaciones numéricas.

    Co-tutela Universitat Politècnica de Catalunya i Université Paul Sabatier

    This thesis focuses on the long time behaviour of solutions to Fisher-KPP reaction-diffusion equations involving fractional diffusion. This type of equation arises, for example, in spatial propagation or spreading of biological species (rats, insects,...). In population dynamics, the quantity under study stands for the density of the population. It is well-known that, under some specific assumptions, the solution tends to a stable state of the evolution problem, as time goes to infinity. In other words, the population invades the medium, which corresponds to the survival of the species, and we want to understand at which speed this invasion takes place. To answer this question, we set up a new method to study the speed of propagation when fractional diffusion is at stake and apply it on three different problems. Part I of the thesis is devoted to an analysis of the asymptotic location of the level sets of the solution to two different problems : Fisher-KPP models in periodic media and cooperative systems, both including fractional diffusion. On the first model, we prove that, under some assumptions on the periodic medium, the solution spreads exponentially fast in time and we find the precise exponent that appears in this exponential speed of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. On the second model, we prove that the speed of propagation is once again exponential in time, with an exponent depending on the smallest index of the fractional Laplacians at stake and on the reaction term. Part II of the thesis deals with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as 'the field' and the line to 'the road', as a reference to the biological situations we have in mind. Indeed, it has long been known that fast diffusion on roads can have a driving effect on the spread of epidemics. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. Contrary to the precise asymptotics obtained in Part I, for this model, we are not able to give a sharp location of the level sets on the road and in the field. The expansion shape of the level sets in the field is investigated through numerical simulations.

    Esta tesis se centra en el comportamiento en tiempos grandes de las soluciones de la ecuación de Fisher- KPP de reacción-difusión con difusión fraccionaria. Este tipo de ecuación surge, por ejemplo, en la propagación espacial o en la propagación de especies biológicas (ratas, insectos,...). En la dinámica de poblaciones, la cantidad que se estudia representa la densidad de la población. Es conocido que, bajo algunas hipótesis específicas, la solución tiende a un estado estable del problema de evolución, cuando el tiempo tiende a infinito. En otras palabras, la población invade el medio, lo que corresponde a la supervivencia de la especie, y nosotros queremos entender con qué velocidad se lleva a cabo esta invasión. Para responder a esta pregunta, hemos creado un nuevo método para estudiar la velocidad de propagación cuando se consideran difusiones fraccionarias, además hemos aplicado este método en tres problemas diferentes. La Parte I de la tesis está dedicada al análisis de la ubicación asintótica de los conjuntos de nivel de la solución de dos problemas diferentes: modelos de Fisher- KPP en medios periódicos y sistemas cooperativos, ambos consideran difusión fraccionaria. En el primer modelo, se prueba que, bajo ciertas hipótesis sobre el medio periódico, la solución se propaga exponencialmente rápido en el tiempo, además encontramos el exponente exacto que aparece en esta velocidad de propagación exponencial. También llevamos a cabo simulaciones numéricas para investigar la dependencia de la velocidad de propagación con la condición inicial. En el segundo modelo, se prueba que la velocidad de propagación es nuevamente exponencial en el tiempo, con un exponente que depende del índice más pequeño de los Laplacianos fraccionarios y también del término de reacción. La Parte II de la tesis ocurre en un entorno de dos dimensiones, donde se reproduce un tipo ecuación de Fisher- KPP con difusión estándar, excepto en una línea del plano, en el que la difusión fraccionada aparece. El plano será llamado "campo" y la línea "camino", como una referencia a las situaciones biológicas que tenemos en mente. De hecho, desde hace tiempo se sabe que la difusión rápida en los caminos puede causar un efecto en la propagación de epidemias. Probamos que la velocidad de propagación es exponencial en el tiempo en el camino, mientras que depende linealmente del tiempo en el campo. Contrariamente a los precisos exponentes obtenidos en la Parte I, para este modelo, no fuimos capaces de dar una localización exacta de los conjuntos de nivel en la carretera y en el campo. La forma de propagación de los conjuntos de nivel en el campo se investiga a través de simulaciones numéricas

  • Elliptic and parabolic PDEs: regularity for nonlocal diffusion equations and two isoperimetric problems

     Serra Montoli, Joaquim
    Universitat Politècnica de Catalunya
    Theses

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    La tesi està dividida en dues parts. La primera part es centra principalment en questions de regularitat per equacions integro - iferencials (o no locals) el·líptiques i parbòliques. De la mateixa manera que les densitats de partícules amb un moviment Brownià resolen equacions el·líptiques o parbòliques de segon ordre, les densitats de partícules amb una difusió de tipus Lévy resolen aquestes equacions no locals més generals. En aquest context, les equacions completament no lineals sorgeixen de problemes de control estocàstic o "differential games''. L'exemple típic d'operador el·liptic no local és el laplacià fraccionari, el qual és l'únic d'aquests operadors que és invariant per translacions, rotacions, i reescalament. Hi ha molts resultats clàssics de regularitat per el laplacià fraccionari --- "l'invers'' del qual és el potencial de Riesz. Per exemple, el nucli de Poisson (explícit) per la bola és un resultat "vell'', així com la teoria de resolubilitat en espais L^p. No obstant això, se sabia ben poc sobre la regularitat a la vora per a aquests problemes. Un tema principal d'aquesta tesi és l'estudi d'aquesta regularitat a la vora , que és qualitativament molt diferent de la de les equacions de segon ordre . A la tesi s'estableix una nova teoria regularitat a la vora per completament no lineals ( i lineals ) equacions integro - diferencials el·líptiques . Les nostres demostracions requeixen una combinació de tècniques originals i versions apropiades de les clàssiques equacions de segon ordre ( com ara el mètode de Krylov ) .    També obtenim nous resultats de regularitat interior per equacions parabòliques no locals completament no lineals i amb "rough kernels''. A tal efecte, desenvolupem un mètode de blow-up i compacitat per a equacions completament no lineals que en permet provar regularitat a partir de teoremes de tipus Liouville. Aquest mètode és una contribució principal de la tesi.   Els nous resultats de regularitat a la vora esmentats anteriorment són essencials en la prova d'un altre resultat principal de la tesi: la identitat Pohozaev per al Laplacià fraccionari. Aquesta identitat recorda a una fórmula d'integració per parts, però amb el Laplacià fraccionari. La novetat important és que apareix un terme de vora locals (això era inusual amb equacions no locals) .      A la segona part de la tesi que donem dos exemples d'interacció entre isoperimetria i Equacions en Derivades Parcials. En el primer, s'utilitza el mètode d'Alexandrov - Bakelman - Pucci per a EDP el · líptiques a fi d'obtenir noves desigualtats isoperimètriques en cons convexos amb densitats, generalitzant una prova de la desigualtat isoperimètric clàssica de X. Cabré . Els nostres nous resultats contenen com a casos particularsla desigualtat clàssica de Wulff i la desigualtat isoperimètrica en cons de Lions-Pacella.    En el segon exemple s'utilitza la desigualtat isoperimètrica i la identitat Pohozaev clàssica per establir un resultat de simetria radial per equacions de reacció - difusió de segon ordre. La novetat en aquest cas és que s'inclouen no - linealitats discontínues . Per a provar aquest resultat, estenem un argument en dues dimensions de P.-L. Lions de 1981 i podem obtenir ara resultass en dimensions superiors.

  • Integro-differential equations: regularity theory and Pohozaev identities

     Ros Oton, Xavier
    Universitat Politècnica de Catalunya
    Theses

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    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

  • Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates

     Cabre Vilagut, Xavier; Sire, Yannick
    Annales de l'Institut Henri Poincaré. Analyse non linéaire
    Vol. 31, num. 1, p. 23-53
    DOI: 10.1016/j.anihpc.2013.02.001
    Date of publication: 2014-01-01
    Journal article

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    This is the first of two articles dealing with the equation (-Delta)(s) upsilon = f (upsilon) in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian - the infinitesimal generator of a Levy process. This equation can be realized as a local linear degenerate elliptic equation in R-+(n+1) together with a nonlinear Neumann boundary condition on partial derivative R-+(n+1) =R-n.; In this first article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian - in the spirit of a result of Modica for the Laplacian. Our proofs are uniform as s up arrow 1, establishing in the limit the corresponding known results for the Laplacian.; In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation. (C) 2013 Elsevier Masson SAS. All rights reserved.

    This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn. In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

  • Sharp energy estimates for nonlinear fractional diffusion equations

     Cabre Vilagut, Xavier; Cinti, Eleonora
    Calculus of variations and partial differential equations
    Vol. 49, num. 1-2, p. 233-269
    DOI: 10.1007/s00526-012-0580-6
    Date of publication: 2014-01
    Journal article

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    We study the nonlinear fractional equation (-Delta)(s) u = f (u) in R-n, for all fractions 0 < s < 1 and all nonlinearities f. For every fractional power s is an element of (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension n = 3 whenever 1/2 <= s < 1. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation -Delta u = f (u) in R-n. It remains open for n = 3 and s < 1/2, and also for n >= 4 and all s.

    We study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0

  • Sobolev and isoperimetric inequalities with monomial weights

     Cabre Vilagut, Xavier; Ros Oton, Xavier
    Journal of differential equations
    Vol. 255, num. 11, p. 4312-4336
    DOI: 10.1016/j.jde.2013.08.010
    Date of publication: 2013
    Journal article

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    We consider the monomial weight |x1|A1¿|xn|An in Rn, where Ai=0 is a real number for each i=1, n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|A1¿|xn|Andx, and they contain the best or critical exponent (which depends on A1, An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions.When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD=RA1+1׿×RAn+1.

    We consider the monomial weight |x1|A1⋯|xn|An in Rn, where Ai⩾0 is a real number for each i=1,…,n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by View the MathML source, and they contain the best or critical exponent (which depends on A1,…,An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD=RA1+1×⋯×RAn+1.

  • Regularity of Stable Solutions up to Dimension 7 in Domains of Double Revolution

     Cabre Vilagut, Xavier; Ros Oton, Xavier
    Communications in partial differential equations
    Vol. 38, num. 1, p. 135-154
    DOI: 10.1080/03605302.2012.697505
    Date of publication: 2013-01
    Journal article

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  • The influence of fractional diffusion in Fisher-KPP equations

     Cabre Vilagut, Xavier; Roquejoffre, Jean-Michel
    Communications in mathematical physics
    Vol. 320, num. 3, p. 679-722
    DOI: 10.1007/s00220-013-1682-5
    Date of publication: 2013
    Journal article

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    We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

  • Ecuaciones en derivadas parciales: problemas de reacción-difusión y problemas geométricos

     Lubary Martinez, Jose Antonio; Gonzalez Nogueras, Maria Del Mar; Haro Cases, Jaime; Sola-morales Rubio, Juan de La Cruz de; Consul Porras, M. Nieves; Cabre Vilagut, Xavier
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  • Fellows of the American Mathematical Society

     Cabre Vilagut, Xavier
    Award or recognition

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  • Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

     Cabre Vilagut, Xavier
    Journal de mathématiques pures et appliquées
    Vol. 98, num. 3, p. 239-256
    DOI: 10.1016/j.matpur.2012.02.006
    Date of publication: 2012-09
    Journal article

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  • Geometric-type Sobolev inequalities and applications to the regularity of minimizers

     Cabre Vilagut, Xavier; Sanchón, Manel
    Journal of functional analysis
    Vol. 264, num. 1, p. 303-325
    DOI: 10.1016/j.jfa.2012.10.012
    Date of publication: 2012-01-01
    Journal article

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  • Euclidean balls solve some isoperimetric problems with nonradial weights

     Cabre Vilagut, Xavier; Ros Oton, Xavier; Serra Montoli, Joaquim
    Comptes rendus de l'Académie des sciences. Série 1, Mathématique
    Vol. 350, num. 21-22, p. 945-947
    DOI: 10.1016/j.crma.2012.10.031
    Date of publication: 2012-11
    Journal article

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  • Propagation in Fisher-KPP type equations with fractional diffusion in periodic media

     Cabre Vilagut, Xavier; Coulon, A.-C.; Roquejoffre, Jean-Michel
    Comptes rendus de l'Académie des sciences. Série 1, Mathématique
    Vol. 350, num. 19-20, p. 885-890
    DOI: 10.1016/j.crma.2012.10.007
    Date of publication: 2012-09
    Journal article

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  • Elliptic problems with axial symmetry: antisymmetry and isoperimetry

     Cabre Vilagut, Xavier
    Progress in Nonlinear Partial Differential Equations
    p. 9
    Presentation's date: 2012-06-05
    Presentation of work at congresses

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  • Jornades d'interacció entre sistemes dinàmics i equacions entre sistemes dinàmics i equacions en derivades parcials (JISD2011)

     Martinez-seara Alonso, Maria Teresa; Delshams i Valdes, Amadeu; Cabre Vilagut, Xavier; Gonzalez Nogueras, Maria Del Mar
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  • Saddle-shaped solutions to the scalar Ginzburg-Landau equation

     Cabre Vilagut, Xavier
    Recent Trends in Nonlinear PDE
    Presentation's date: 2011-09-22
    Presentation of work at congresses

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  • Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

     Cabre Vilagut, Xavier
    Geometric properties of solutions of nonlinear PDEs and their applications
    Presentation's date: 2011-07-17
    Presentation of work at congresses

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  • Qualitative properties of saddle-shaped solutions to the Allen-Cahn equation

     Cabre Vilagut, Xavier
    Escola Brasileira de Equaçoes Diferenciais
    Presentation's date: 2011-08
    Presentation of work at congresses

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  • Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

     Cabre Vilagut, Xavier
    Nonlinear PDEs and Functional Inequalities
    Presentation's date: 2011-09-19
    Presentation of work at congresses

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  • Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

     Cabre Vilagut, Xavier
    Fronts et EDP non linéaires: colloque en l'honneur de Henri Berestycki
    Presentation's date: 2011-06-20
    Presentation of work at congresses

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  • EQUACIONS DIFERENCIALS EN DERIVADES PARCIALS I APLICACIONS

     Valencia Guitart, Marta; Consul Porras, M. Nieves; Aguareles Carrero, Maria; Lubary Martinez, Jose Antonio; Sola-morales Rubio, Juan de La Cruz de; Mande Nieto, Jose Vicente; Gonzalez Nogueras, Maria Del Mar; Lucia D'Agostino, Marcello; Haro Cases, Jaime; Cabre Vilagut, Xavier
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  • Jornades d'introducció als sistemes dinàmics i a les EDP's 2010 (JISD2010)

     Gonzalez Nogueras, Maria Del Mar; Martinez-seara Alonso, Maria Teresa; Cabre Vilagut, Xavier
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  • Positive solutions of nonlinear problems involving the square root of the Laplacian

     Cabre Vilagut, Xavier; Tan, Jinggang
    Advances in mathematics
    Vol. 224, num. 5, p. 2052-2093
    DOI: 10.1016/j.aim.2010.01.025
    Date of publication: 2010-08-01
    Journal article

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  • Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

     Cabre Vilagut, Xavier; Terra, J.
    Communications in partial differential equations
    Vol. 35, num. 11, p. 19231-1957
    DOI: 10.1080/03605302.2010.484039
    Date of publication: 2010
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  • Regularity of minimizers of semilinear elliptic problems up to dimension 4

     Cabre Vilagut, Xavier
    Communications on pure and applied mathematics
    Vol. 63, num. 10, p. 1362-1380
    DOI: 10.1002/cpa.20327
    Date of publication: 2010-10
    Journal article

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  • Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

     Cabre Vilagut, Xavier; Cinti, Eleonora
    Discrete and continuous dynamical systems. Series A
    Vol. 28, num. 3, p. 1179-1206
    DOI: 10.3934/dcds.2010.28.1179
    Date of publication: 2010-11
    Journal article

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    We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every nonlinearity and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension , we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation in R n.

  • Bistable elliptic equations with fractional diffusion

     Cinti, Eleonora
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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  • Phase transitions and front propagation for fractional diffusion equations

     Cabre Vilagut, Xavier
    Nonlinear evolution equations
    Presentation's date: 2010-06
    Presentation of work at congresses

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  • Front propagation and phase transitions for fractional diffusion equations

     Cabre Vilagut, Xavier
    Joint SIAM/RSME-SCM-SEMA Meeting
    Presentation's date: 2010-06-04
    Presentation of work at congresses

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  • Liouville theorems in bistable diffusion equations. Standard and fractional diffusion (I - II)

     Cabre Vilagut, Xavier
    Nonlinear PDE Days
    p. 1
    Presentation's date: 2010-01
    Presentation of work at congresses

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  • ECUACIONES EN DERIVADAS PARCIALES; ANÁLISIS Y APLICACIONES

     Valencia Guitart, Marta; Consul Porras, M. Nieves; Aguareles Carrero, Maria; Lubary Martinez, Jose Antonio; Sola-morales Rubio, Juan de La Cruz de; Mande Nieto, Jose Vicente; Gonzalez Nogueras, Maria Del Mar; Lucia D'Agostino, Marcello; Haro Cases, Jaime; Cabre Vilagut, Xavier
    Competitive project

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  • Saddle-shaped solutions of bistable diffusion equations in all of R2m

     Cabre Vilagut, Xavier; Mourao Terra, Joana
    Journal of the European Mathematical Society
    Vol. 11, num. 4, p. 819-843
    DOI: 10.4171/JEMS/168
    Date of publication: 2009
    Journal article

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    We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation 1u D f .u/ in the whole R2m, where f is of bistable type. It is known that in dimension 2m D 2 there exists a saddle-shaped solution. This is a solution which changes sign in R2 and vanishes only on fjx1j D jx2jg. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2m D 4. More precisely, our main result establishes that if 2m D 4, every solution vanishing on the Simons cone f.x1; x2/ 2 Rm Rm : jx1j D jx2jg is unstable outside every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.

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    Positive solutions of nonlinear problems involving the square root of the Laplacian  Open access

     Cabre Vilagut, Xavier; Tan, Jinggang
    Date: 2009-05
    Report

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    We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

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    Front propagation in Fisher-KPP equations with fractional diffusion  Open access

     Cabre Vilagut, Xavier; Roquejoffre, Jean-Michel
    Date: 2009-05
    Report

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    We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.

  • Front propagation in Fisher-KPP equations with fractional diffusion

     Cabre Vilagut, Xavier; Roquejoffre, Jean-Michel
    Comptes rendus de l'Académie des sciences. Série 1, Mathématique
    Vol. 347, num. 23-24, p. 1361-1366
    DOI: 10.1016/j.crma.2009.10.012
    Date of publication: 2009-12
    Journal article

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    Front propagation in Fisher–KPP equations with fractional diffusion.We study in this Note the Fisher–KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.

  • Regularity of radial minimizers of reaction equations involving the p-Laplacian

     Cabre Vilagut, Xavier; Capella Kort, Antonio; Sanchón, Manel
    Calculus of variations and partial differential equations
    Vol. 34, num. 4, p. 475-494
    Date of publication: 2009-04
    Journal article

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  • Saddle-shaped solutions of bistable diffusion equations in all of $\Bbb R^{2m}$

     Cabre Vilagut, Xavier
    Congreso de la Real Sociedad de Matemática Española 2009
    p. 7
    Presentation's date: 2009-02-06
    Presentation of work at congresses

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  • NÁLISIS NO LINEAL PARA UN RETÍCULO ELÁSTICO Y PARA UN LAPLACIANO FRACCIONARIO

     TAN, JINGGANG
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
    Theses

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  • Nonlinear elliptic PDE¿s: stability properties of saddle-shaped and minimal solutions

     TERRA, JOANA
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
    Theses

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  • A priori bounds for stable solutions of nonlinear elliptic equations in low dimensions

     Cabre Vilagut, Xavier
    ICMC Summer Meeting on Differential Equations - 2008 Chapter
    Presentation's date: 2008-01-28
    Presentation of work at congresses

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  • Saddle-shaped solutions of bistable diffusion equations

     Cabre Vilagut, Xavier
    8th International Conference on Operations Research, Session ?Ecuaciones en Derivadas Parciales y Aplicaciones?
    Presentation's date: 2008-02-24
    Presentation of work at congresses

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  • Nonlinear Diffusion PDE¿s: Transition Layers and Front Propagation

     Cabre Vilagut, Xavier
    NoLineal 2008
    Presentation's date: 2008-06-17
    Presentation of work at congresses

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  • Saddle-shaped solutions of bistable difusion equations in all of R^2n

     Cabre Vilagut, Xavier
    Workshop on Variational Methods for Nonlinear PDE and their Applications
    Presentation of work at congresses

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  • A priori bounds for stable solutions of nonlinear elliptic equations in low dimensions

     Cabre Vilagut, Xavier
    ICMC Summer Meeting on Diferential Equations - 2008 Chapter
    Presentation's date: 2008-01-01
    Presentation of work at congresses

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  • Saddle-shaped solutions of bistable diffusion equations in all of R^{2m}

     Cabre Vilagut, Xavier; Mourao Terra, Joana
    Date: 2008-01
    Report

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  • Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions

     Cabre Vilagut, Xavier
    Discrete and continuous dynamical systems. Series A
    Vol. 20, num. 3, p. 425-457
    Date of publication: 2008-03
    Journal article

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  • A priori bounds for stable solutions of nonlinear elliptic equations in low dimensions

     Cabre Vilagut, Xavier
    ICMC Summer Meeting on Differential Equations - 2008 Chapter
    Presentation of work at congresses

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  • Saddle-Shaped Solutions of Bistable Diffusion Equations in all of R^2M

     Cabre Vilagut, Xavier
    NoLineal 2008
    p. 27-28
    Presentation of work at congresses

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  • Saddle-shaped solutions of bistable difusion equations in all of R 2n

     Cabre Vilagut, Xavier
    Workshop on Variational Methods for Nonlinear PDE and their Applications
    Presentation's date: 2008-03-01
    Presentation of work at congresses

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