Browsing by Author "Benguria, Rafael D."

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    A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities
    (2021) Antunes, Pedro R. S.; Benguria, Rafael D.; Lotoreichik, Vladimir; Ourmieres-Bonafos, Thomas
    We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of R-2. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szego type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.
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    AN ITERATIVE ESTIMATION FOR DISTURBANCES OF SEMI-WAVEFRONTS TO THE DELAYED FISHER-KPP EQUATION
    (2019) Benguria, Rafael D.; Solar, Abraham
    We give an iterative method to estimate the disturbance of semi-wavefronts of the equation (u) over dot (t, x) = u '' (t, x) + u(t, x)(1 - u(t - h, x)), x is an element of R, t > 0, where h > 0. As a consequence, we show the exponential stability, with an unbounded weight, of semi- wavefronts with speed c >= 2 v 2 and h > 0. Under the same restriction of c and h, the uniqueness of semi- wavefronts is obtained.
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    Gagliardo-Nirenberg-Sobolev inequalities for convex domains in Rd
    (2019) Benguria, Rafael D.; Vallejos, Cristobal; Van Den Bosch, Hanne
    A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in R-d has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of R-d, in particular for cubes, has arisen. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.
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    Remarks on the spectrum of a non-local Dirichlet problem
    (2021) Benguria, Rafael D.; Pereira, Marcone C.
    In this paper, we analyse the spectrum of non-local Dirichlet problems with non-singular kernels in bounded open sets. The novelty is twofold. First we study the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Next, under additional smooth conditions on the kernel and domain, we prove differentiability of simple eigenvalues computing their first derivative discussing extremum problems for eigenvalues.
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    Variational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equation
    (2022) Benguria, Rafael D.; Depassier, M. Cristina; Rica, Sergio
    We examine non-linear diffusive front propagation in the frame of the Fisher-type equation: dtu = dx (D(u)dxu)+ u(1 - u). We study the problem of a sudden jump in diffusivity motivated by models of glassy polymers. It is shown that this problem differs substantially from the problem of front propagation in the usual Fisher equation which was solved by Kolmogorov, Petrovsky, and Piskunov (KPP) in 1937. As in the Fisher, Kolmogorov, Petrovsky, Piskunov (FKPP) problem, the asymptotic dynamics of the non linear diffusive front propagation is reduced to the study of a nonlinear ordinary differential equation with adequate boundary conditions. Since this problem does not allow an exact result for the propagation speed, we use a variational approach to estimate the front speed and compare it with direct time-dependent numerical simulations showing an excellent agreement.