We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. / text

In this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is due to the appearance of the `fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular (Gaussian) diffusion. Motivated by the observation that, in contrast to the classical Fokker-Planck equation (describing regular diffusion in the presence of a potential field), there is no natural gradient flow formulation for its fractional counterpart, we prove existence of weak solutions to this fractional Fokker-Planck equation by combining a splitting technique together with a Wasserstein gradient flow formulation. An explicit iterative construction is given, which we prove weakly converges to a weak solution of this PDE. / Graduate

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

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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 $alpha+2$ 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 $0 <alpha < 1$ et $1 leq alpha < 2$. 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 $0 <alpha < 1$ et $1 leq alpha < 2$ 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 solutions / 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 property

Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.

The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

The presented work deals with two distinct problems in the field of Mathematical Physics. The first part is dedicated to an 'almost' solution of Villani's conjecture, a known conjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzmann equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equilibrate is proportional to the number of particles in the system. Our main result in this part is a proof, up to an epsilon, of that conjecture, showing that for all practical purposes we can consider it to be true. The second part of the presentation is based on a joint work with Prof. Michael Loss and is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to intersection of hyperplanes. The newly found inequality is sharp and the functions that attain equality in it are completely classified.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-14T16:13:37Z No. of bitstreams: 1 arquivototal.pdf: 1041120 bytes, checksum: 3357ded46458082b719eebe4f03879a9 (MD5) / Made available in DSpace on 2017-08-14T16:13:37Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1041120 bytes, checksum: 3357ded46458082b719eebe4f03879a9 (MD5) Previous issue date: 2015-12-18 / In this work we prove some results of existence and multiplicity of solutions for equations of the type (􀀀 ) u + V (x)u = f(x; u) in RN; where 0 < < 1, N 2 , (􀀀 ) denotes the fractional Laplacian, V : RN ! R is a continuous function that satisfy suitable conditions and f : RN R ! R is a continuous function that may have critical growth in the sense of the Trudinger-Moser inequality or in the sense of the critical Sobolev exponent. In order to obtain our results we use variational methods combined with a version of the Concentration-Compactness Principle due to Lions. / Neste trabalho provamos alguns resultados de existência e multiplicidade de soluções para equações do tipo (􀀀 ) u + V (x)u = f(x; u) em RN; onde 0 < < 1, N 2 , (􀀀 ) denota o Laplaciano fracionário, V : RN ! R é uma função contínua que satisfaz adequadas condições e f : RN R ! R é uma função cont ínua que pode ter crescimento crítico no sentido da desigualdade de Trudinger-Moser ou no sentido do expoente crítico de Sobolev. A m de obter nossos resultados usamos métodos variacionais combinados com uma versão do Princípio de Concentração- Compacidade devido à Lions.

Orientadores: Lucas Catão de Freitas Ferreira, Marcelo da Silva Montenegro, José Antonio Carrillo de la Plata / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T15:13:15Z (GMT). No. of bitstreams: 1 Santos_MatheusCorreiados_D.pdf: 865476 bytes, checksum: 31a8b558231b701d81c20bf2712e4f50 (MD5) Previous issue date: 2015 / Resumo: Analisaremos dois problemas neste trabalho. Na primeira parte, estudaremos a existência de soluções para uma equação elíptica semilinear no espaço euclidiano todo e com dependência do gradiente e onde nenhuma restrição é imposta sobre o comportamento da não linearidade no infinito. Provaremos que existe uma solução que é localmente única e que herda muitas das propriedades de simetria da não linearidade. A positividade da solução e seu comportamento assintótico também são analisados. Os resultados obtidos também podem ser estendidos para outros casos como o de domínios exteriores ou o semiespaço e também para alguns operadores fracionários. Na segunda parte, analisaremos o comportamento assintótico das soluções da versão fracionária unidimensional da equações de meios porosos introduzida por Caffarelli e Vázquez e onde a pressão é obtida como a inversa do laplaciano fracionário da densidade. Devido à convexidade do núcleo do potencial de Riesz em dimensão um, mostraremos que a entropia associada à equação é displacement convex e satisfaz uma desigualdade funcional envolvendo a dissipação da entropia e a distância de transporte euclidiana. Um argumento por aproximação mostra que essa desigualdade funcional é suficiente para deduzir que a entropia das soluções converge exponencialmente para a entropia do estado estacionário. Também provaremos uma nova desigualdade de interpolação que permitirá obter a convergência exponencial das soluções em espaços Lp / Abstract: We analyse two problems in this work. In the first part we study the existence of solutions to a semilinear elliptic equation in the whole space and with dependence on the gradient and where no restriction is imposed on the behavior of the nonlinearity at infinity. We prove that there exists a solution which is locally unique and inherits many of the symmetry properties of the nonlinearity. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains like half-space and exterior domains and also to some fractional operators. For the second part, we analyse the asymptotic behavior of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vázquez and where the pressure is obtained as the inverse of the fractional Laplacian of the density. Due to the convexity of the kernel of the Riesz potential in one dimension, we show that the entropy associated with the equation is displacement convex and satisfies a functional inequality involving also entropy dissipation and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence, in the entropy level, of solutions to the unique steady state. A new interpolation inequality is also proved in order to obtain the exponential decay also in Lp spaces / Doutorado / Matematica / Doutor em Matemática