A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev-Poincar\'e inequality for variable exponent spaces by means of mass transportation methods. The importance of this approach is that the method is exible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result obtained by Fan.Comment: 12 page

Finite element approximation of fractional Neumann problems

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and asymptotic behavior of solutions. We address the convergence of the finite element discretizations and discuss the implementation of the method. Finally, we present several numerical experiments in one- and two-dimensional domains that illustrate the method's performance as well as certain properties of solutions

Finite element approximation for the fractional eigenvalue problem

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.Comment: 20 pages, 6 figure

Constructive approximation on graded meshes for the integral fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian. We prove optimal Sobolev regularity estimates in Lipschitz domains satisfying an exterior ball condition. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability

Flujo de fluídos estratificados.

El flujo de un fluído estratificado en un canal abierto presenta un comportamiento más complejo que el de una capa homogénea. Una de las mayores dificultades para el estudio de este fenómeno es la ausencia de un único parámetro adimensionalizado (número de Froude) que caracterice al flujo. Es usual simplificar el problema considerando una estratificación discreta, suponiendo que el sistema está formado por dos o más estratos de densidad y velocidad uniforme. La complejidad del flujo aumenta considerablemente con la cantidad de capas. Resulta relevante considerar la posibilidad de caracterizar cualitiva y cuantitativamente un flujo de esta complejidad mediante una o más capas activas. En este trabajo, se estudia bajo qué condiciones y hasta qué punto es viable simplificar el flujo en base a consideraciones de dicho tipo. Los resultados analíticos son contrastados con los obtenidos mediante simulaciones numéricas propias

Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $\Omega$: $$\|u\|_{\dot{B}^{s+r}_{2,\infty}(\Omega)} \le C \|f\|_{L^2(\Omega)} \quad r = \min\{s,1/2\}.$$ This estimate is consistent with the regularity on smooth domains and shows that there is no loss of regularity due to Lipschitz boundaries. The proof uses elementary ingredients, such as the variational structure of the problem and the difference quotient technique

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight $w$ times a ``regular´´ unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the ``regular´´ unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babu{s}ka and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nystr"om numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Borthagaray, Juan Pablo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Bruno, Oscar Ricardo. California Institute Of Technology; Estados UnidosFil: Maas, Martín Daniel. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentin