2012
, Option Pricing for Stocks with Dividends: An Analytic Approach by PDEs, Monografıas de la Real Academia de Ciencias de Zaragoza (38), pp. 125-136. (2012). 
, Positivity and negativity of solutions to nxn weighted systems involving the Laplace operator defined on $R^n$, $n >=3$, Electronic Journal of Differential Equations (101), pp. 1-14. (2012).
2009
, Intrinsic ultracontractivity of a Schrödinger semigroup in $mathbb R^N$, Journal of Functional Analysis, 256 (12), pp. 4095--4127. (2009).
, Principal eigenvalue and maximum principle for some elliptic systems defined on general domains with refined Dirichlet boundary condition, Communications in Mathematical Analysis, 7 (2), pp. 1--11. (2009).
We prove the existence of a principal eigenvalue and we derive a Refined Maximum Principle for an elliptic system $LU = \Lambda MU +F$ defined on an irregular bounded domain in $\mathbb R^N$ with Refined Dirichlet Boundary Condition; here $L$ is a diagonal matrix of uniformly elliptic operators $L_i$; $1 \leq i \leq n$; $F$ is a vector with bounded, positive entries; $M$ is a cooperative $nxn$ matrix, with terms on the diagonal allowed to change sign. This extends some of the results in Beresticky, Niremberg, Varadhan ; Birindelli ; Birindelli, Mitidieri, Sweers ; Fleckinger, Hernandez, de Thélin.
We prove the existence of a principal eigenvalue and we derive a Refined Maximum Principle for an elliptic system $LU = \Lambda MU +F$ defined on an irregular bounded domain in $\mathbb R^N$ with Refined Dirichlet Boundary Condition; here $L$ is a diagonal matrix of uniformly elliptic operators $L_i$; $1 \leq i \leq n$; $F$ is a vector with bounded, positive entries; $M$ is a cooperative $nxn$ matrix, with terms on the diagonal allowed to change sign. This extends some of the results in Beresticky, Niremberg, Varadhan ; Birindelli ; Birindelli, Mitidieri, Sweers ; Fleckinger, Hernandez, de Thélin.
2007
, Compactness for a Schrödinger Operator in the Ground State Space over $\mathbb{R}^n$, Electronic Journal of Differential Equations, Proceedings of the 2006 International Conference in honor of Jacqueline Fleckinger, 16, pp. 35- - 58. (2007).
, Ground-state positivity, negativity, and compactness for a Schrödinger operator in $\mathbb{R}^N$, Journal of functional analysis, 245 (1), pp. 213-248. (2007).
2006
, Systems of Schrödinger equations : positivity and negativity, Monografias des Seminario Matematico Garcia de Galdeano, Ninth International Conference Zaragoza-Pau on Applied Mathematics and Statistics, 33, pp. 19--26. (2006).
We consider here Schrödinger operator $-\Delta+q(x)$ defined in the entire space $\mathbb R^N$, with a potential $q$ tending to $+\infty$ at infinity with a sufficiently fast growth. The ground state positivity and negativity for a Schrödinger equation with spectral parameter says that, if the spectral parameter is lower that the principal eigenvalue, the solutions satisfy ground state positivity (greater than a positive constant times the ground state) and if the spectral parameter is slightly greater than the principal eigenvalue, then the solutions satisfy ground state negativity (lower than minus a positive constant times the ground state). We extend this ground state positivity and negativity to cooperative and noncooperative systems of two Schrödinger equations.
We consider here Schrödinger operator $-\Delta+q(x)$ defined in the entire space $\mathbb R^N$, with a potential $q$ tending to $+\infty$ at infinity with a sufficiently fast growth. The ground state positivity and negativity for a Schrödinger equation with spectral parameter says that, if the spectral parameter is lower that the principal eigenvalue, the solutions satisfy ground state positivity (greater than a positive constant times the ground state) and if the spectral parameter is slightly greater than the principal eigenvalue, then the solutions satisfy ground state negativity (lower than minus a positive constant times the ground state). We extend this ground state positivity and negativity to cooperative and noncooperative systems of two Schrödinger equations.
2005
, Anti-maximum principle for a Schrödinger equation in $\mathbb R^N$, with a non radial potential, Rostocker Mathematisches Kolloquium (59), pp. 51--62. (2005).
2004
, Variational methods for a resonant problem with the $p$-Laplacian in $\mathbb R^N$, Electronic Journal of Differential Equations, 76, pp. 32. (2004).
, On some extended maximum and antimaximum principles, VIII Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques, 31, pp. 13--20. (2004).
2003
, Anti-maximum principle for cooperative system involving Schrödinger operator in ${\mathbb R}^N$, Seventh Zaragoza-Pau Conference on Applied Mathematics and Statistics (Spanish) (Jaca, 2001), 27, pp. 37--40. (2003).
, Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in $\mathbb R^2$, Mathematical Methods in the Applied Sciences, 26 (13), pp. 1093--1136. (2003).
2002
, Maximum and antimaximum principles for some elliptic systems involving Schrödinger operators, Partial differential equations, Lecture Notes in Pure and Appl. Math., 229, pp. 13--30. (2002).
2001
, Positivity and negativity of solutions to a Schrödinger equation in $\mathbb R^N$, Positivity. An International Journal devoted to the Theory and Applications of Positivity in Analysis, 5 (4), pp. 359--382. (2001).
1999
, Maximum and anti-maximum principles for some systems involving Schrödinger operators, Oper. Theory Adv. Appl., The Mazya anniversary collection, Vol. 2 (Rostock, 1998), 110, pp. 13--21. (1999).
, An extension of maximum and anti-maximum principles to a Schrödinger in $\mathbb R^2$, Journal of Differential Equations, 156 (1), pp. 122--152. (1999).
1997
, A pointwise lower bound for positive solutions of a Schrödinger equation in $\mathbb R^N$, Journal of Differential Equations, 133 (2), pp. 280--295. (1997).
, Maximum principle and existence of solutions for elliptic systems involving Schrödinger operators, Revista de la Real Academia de Ciencias Exactas, F\'\i sicas y Naturales (Espa\ na), 91 (1), pp. 47--52. (1997).
, A PDE Approach to Asian Options : analytical and numerical evidence, Journal of Banking and Finance, 21, pp. 613--640. (1997).
1994
, A grid refinement method for deterministic control and differential games, Mathematical Models & Methods in Applied Sciences, 4 (6), pp. 899--910. (1994).
1991
, Jeux différentiels et approximation numérique de fonctions valeur. II. étude numérique, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (5), pp. 535--560. (1991).
, Jeux différentiels et approximation numérique de fonctions valeur. I. étude théorique, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (5), pp. 517--533. (1991).
