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Tarhini Rana - One of the best experts on this subject based on the ideXlab platform.
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Équation De films minces fractionnaire pour les fractures hydrauliques
HAL CCSD, 2018Co-Authors: Tarhini RanaAbstract:In this thesis, we study two Degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove HölDer regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local Degenerate parabolic equation of orDer $alpha + 2$ where $0 < alpha < 2$. It is a generalization of an equation studied by Imbert and Mellet in 2011 for $alpha = 1$. To construct these solutions, we consiDer a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$. We also prove that the solution is positive if the initial condition is so. The fourth chapter is Dedicated for a fractional porous medium equation. We prove HölDer regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$ because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove HölDer regularity of solutions using the scaling propertyCes travaux concernent Deux équations paraboliques, dégénérées et non-locales. La première équation est une équation De films minces fractionnaire et la Deuxième est une équation Des milieux poreux fractionnaire. La présentation Des problèmes, les résultats existants dans la littérature, ainsi que le résumé De nos résultats font l'objet De l'introduction. Le Deuxième chapitre est consacré à la présentation De la méthoDe De De Giorgi utilisée pour montrer la régularité HölDer Des solutions Des équations elliptiques. On présente De plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence De solutions faibles d'une équation Des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre $alpha+2$ où $0 < alpha < 2$. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour $alpha = 1$. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections De Sobolev, on passe à la limite pour trouver Des solutions faibles. Vu la différence Des injections De Sobolev, on distingue Deux cas $0
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Fractional equation of thin films for hydraulic fractures
2018Co-Authors: Tarhini RanaAbstract:Ces travaux concernent Deux équations paraboliques, dégénérées et non-locales. La première équation est une équation De films minces fractionnaire et la Deuxième est une équation Des milieux poreux fractionnaire. La présentation Des problèmes, les résultats existants dans la littérature, ainsi que le résumé De nos résultats font l'objet De l'introduction. Le Deuxième chapitre est consacré à la présentation De la méthoDe De De Giorgi utilisée pour montrer la régularité HölDer Des solutions Des équations elliptiques. On présente De plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence De solutions faibles d'une équation Des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre α+2 où 0 ≺ α ≺ 2. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour α = 1. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections De Sobolev, on passe à la limite pour trouver Des solutions faibles. Vu la différence Des injections De Sobolev, on distingue Deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2. Dans les Deux cas on démontre que la solution est positive si la condition initiale l'est. Le quatrième chapitre concerne une équation Des milieux poreux fractionnaire. On montre la régularité HölDer De solutions faibles positives satisfaisant Des estimées d'énergie. D'abord, on montre l'existence De solutions faibles qui satisfont Des estimées d'énergie. On distingue Deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2 à cause De problème De divergence. Puis on démontre les lemmes De De Giorgi qui sont Des lemmes De réduction De l'oscillation d'en Dessus et d'au-Dessous. Ces Deux lemmes ne suffisent pas pour montrer la régularité HölDer. On a besoin d'améliorer le résultat du lemme De réduction De l'oscillation d'en Dessus. Donc, on passe par un lemme Des valeurs intermédiaires et on montrer un lemme De réduction De l'oscillation d'en Dessus amélioré. Enfin, on montre la régularité HölDer Des solutions en utilisant la propriété scaling De ces solutionsIn this thesis, we study two Degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove HölDer regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local Degenerate parabolic equation of orDer α+2 where 0 ≺ α ≺ 2. It is a generalization of an equation studied by Imbert and Mellet in 2011 for α = 1. To construct these solutions, we consiDer a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases 0 ≺ α ≺ 1 and 1 ≤ α ≺ 2. We also prove that the solution is positive if the initial condition is so. The fourth chapter is Dedicated for a fractional porous medium equation. We prove HölDer regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases 0 ≺ α ≺ 1 and 1 ≤ α ≺ 2 because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove HölDer regularity of solutions using the scaling propert
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Équation De films minces fractionnaire pour les fractures hydrauliques
HAL CCSD, 2018Co-Authors: Tarhini RanaAbstract:In this thesis, we study two Degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove HölDer regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local Degenerate parabolic equation of orDer α+2 where 0 ≺ α ≺ 2. It is a generalization of an equation studied by Imbert and Mellet in 2011 for α = 1. To construct these solutions, we consiDer a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases 0 ≺ α ≺ 1 and 1 ≤ α ≺ 2. We also prove that the solution is positive if the initial condition is so. The fourth chapter is Dedicated for a fractional porous medium equation. We prove HölDer regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases 0 ≺ α ≺ 1 and 1 ≤ α ≺ 2 because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove HölDer regularity of solutions using the scaling propertyCes travaux concernent Deux équations paraboliques, dégénérées et non-locales. La première équation est une équation De films minces fractionnaire et la Deuxième est une équation Des milieux poreux fractionnaire. La présentation Des problèmes, les résultats existants dans la littérature, ainsi que le résumé De nos résultats font l'objet De l'introduction. Le Deuxième chapitre est consacré à la présentation De la méthoDe De De Giorgi utilisée pour montrer la régularité HölDer Des solutions Des équations elliptiques. On présente De plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence De solutions faibles d'une équation Des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre α+2 où 0 ≺ α ≺ 2. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour α = 1. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections De Sobolev, on passe à la limite pour trouver Des solutions faibles. Vu la différence Des injections De Sobolev, on distingue Deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2. Dans les Deux cas on démontre que la solution est positive si la condition initiale l'est. Le quatrième chapitre concerne une équation Des milieux poreux fractionnaire. On montre la régularité HölDer De solutions faibles positives satisfaisant Des estimées d'énergie. D'abord, on montre l'existence De solutions faibles qui satisfont Des estimées d'énergie. On distingue Deux cas 0 ≺ α ≺ 1 et 1 ≤ α ≺ 2 à cause De problème De divergence. Puis on démontre les lemmes De De Giorgi qui sont Des lemmes De réduction De l'oscillation d'en Dessus et d'au-Dessous. Ces Deux lemmes ne suffisent pas pour montrer la régularité HölDer. On a besoin d'améliorer le résultat du lemme De réduction De l'oscillation d'en Dessus. Donc, on passe par un lemme Des valeurs intermédiaires et on montrer un lemme De réduction De l'oscillation d'en Dessus amélioré. Enfin, on montre la régularité HölDer Des solutions en utilisant la propriété scaling De ces solution
Chunyi Zhao - One of the best experts on this subject based on the ideXlab platform.
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on phase separation moDels asymptotics and qualitative properties
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Henri Berestycki, Taichia Lin, Juncheng Wei, Chunyi ZhaoAbstract:In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains diviDed by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounDed solutions with symmetry and monotonicity. These unbounDed solutions are useful for rigorously Deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of Du and Zhang (Discontin Dynam Sys, 2012). When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system unDer suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for these types of systems in higher dimensions.
Enrico Valdinoci - One of the best experts on this subject based on the ideXlab platform.
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a three dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime
arXiv: Analysis of PDEs, 2017Co-Authors: Serena Dipierro, Enrico Valdinoci, Alberto FarinaAbstract:We consiDer bounDed solutions of the nonlocal Allen-Cahn equation $$ (-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ unDer the monotonicity condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in which~$s\in\left(0,\frac12\right)$. UnDer the limit assumptions $$ \lim_{x_n\to-\infty} u(x',x_n)=-1\quad{\mbox{ and }}\quad \lim_{x_n\to+\infty} u(x',x_n)=1,$$ it has been recently shown that~$u$ is necessarily $1$D, i.e. it Depends only on one EucliDean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by Ennio De Giorgi.
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regularity and bernstein type results for nonlocal minimal surfaces
arXiv: Analysis of PDEs, 2013Co-Authors: Alessio Figalli, Enrico ValdinociAbstract:We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$.
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bernstein and De Giorgi type problems new results via a geometric approach
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 2009Co-Authors: Alberto Farina, Berardino Sciunzi, Enrico ValdinociAbstract:We use a Poincare type formula and level set analysis to Detect one-dimensional symmetry of stable solutions of possibly Degenerate or singular elliptic equation of the form div a(|∇u(x)|)∇u(x) + f(u(x)) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in R and R and of the Bernstein problem on the flatness of minimal area graphs in R. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to non-elliptic operators: as an application, we prove one-dimensional symmetry for 1-Laplacian type operators.
Luigi Ambrosio - One of the best experts on this subject based on the ideXlab platform.
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gamma convergence of nonlocal perimeter functionals
arXiv: Functional Analysis, 2010Co-Authors: Luigi Ambrosio, Guido De Philippis, Luca MartinazziAbstract:We prove that certain non-local functionals Defined on measurable sets Gamma-converge to the perimeter in the sense of De Giorgi.
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locality of the perimeter in carnot groups and chain rule
Annali di Matematica Pura ed Applicata, 2010Co-Authors: Luigi Ambrosio, Matteo ScienzaAbstract:In the class of Carnot groups, we study fine properties of sets of finite perimeter. Improving a recent result by Ambrosio–Kleiner–Le Donne, we show that the perimeter measure is local, i.e., that given any pair of sets of finite perimeter their perimeter measures coinciDe on the intersection of their essential boundaries. This solves a question left open in Ambrosio et al. (Calculus of variations: topics from mathematical heritage of Ennio De Giorgi. Quad Mat). As a consequence, we prove a general chain rule for BV functions in this setting.
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On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property
Acta Applicandae Mathematica, 2001Co-Authors: Giovanni Alberti, Luigi Ambrosio, Xavier CabréAbstract:This paper studies a conjecture maDe by De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounDed, monotone in one direction, solutions of semilinear elliptic equations Δ u = F ′( u ) in all of R ^ n . We extend to all nonlinearities F ∈ C ^2 the symmetry result in dimension n =3 previously established by the second and third authors for a class of nonlinearities F which incluDed the moDel case F ′( u )= u ^3− u . The extension of the present paper is based on new energy estimates which follow from a local minimality property of u . In addition, we prove a symmetry result for semilinear equations in the halfspace R _+ ^4. Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n ≤8, namely that the level sets of u are flat at infinity.
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on a long standing conjecture of e De Giorgi symmetry in 3d for general nonlinearities and a local minimality property
Acta Applicandae Mathematicae, 2001Co-Authors: Giovanni Alberti, Luigi Ambrosio, Xavier CabréAbstract:This paper studies a conjecture maDe by E. De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounDed, monotone in one direction, solutions of semilinear elliptic equations u = F 0 (u) in all of R n . We extend to all nonlinearities F 2 C 2 the symmetry result in dimension n = 3 previously established by the second and the third authors for a class of nonlinearities F which incluDed the moDel case F 0 (u) = u 3 u. The extension of the present paper is based on a new energy estimates which follow from a local minimality property of u. In addition, we prove a symmetry result for semilinear equations in the halfspace R 4. Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n 8, namely that the level sets of u are at at innity.
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entire solutions of semilinear elliptic equations in r 3 and a conjecture of De Giorgi
Journal of the American Mathematical Society, 2000Co-Authors: Luigi AmbrosioAbstract:This paper is concerned with the study of bounDed solutions of semilinear elliptic equations u F u in the whole space R unDer the assumption that u is monotone in one direction say nu in R n The goal is to establish the one dimensional character or symmetry of u namely that u only Depends on one variable or equivalently that the level sets of u are hyperplanes This type of symmetry question was raised by De Giorgi in who maDe the following conjecture we quote literally page of DG
Paolo Tilli - One of the best experts on this subject based on the ideXlab platform.
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nonlinear wave equations as limits of convex minimization problems proof of a conjecture by De Giorgi
Annals of Mathematics, 2012Co-Authors: Enrico Serra, Paolo TilliAbstract:We prove a conjecture by De Giorgi, which states that global weak solutions of nonlinear wave equations such as w + jwj p 2 w = 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations. These functionals, which are integrals in space-time of a convex Lagrangian, contain an exponential weight with a parameter ", and the initial data of the wave equation serve as boundary conditions. As " tends to zero, the minimizers v" converge, up to subsequences, to a solution of the nonlinear wave equation. There is no restriction on the nonlinearity exponent, and the method is easily extenDed to more general equations.