Members   Juan Diego Dávila Bonczos

Juan Diego Dávila Bonczos


Ph.D. Rutgers University, 2002
Associate Professor
Departamento de Ingeniería Matemática y
Centro de Modelamiento Matemático
Universidad de Chile


Research interests

I am interested in nonlinear partial differential equations of elliptic and parabolic type. One line of research is connected to the Gelfand-Liouville equation, which appears in geometry and models of combustion and equilibrium of stars. The general goal is to describe the bifurcation diagram for the solution set, and in this analysis singular solutions appear naturally. I am interested in constructing these solutions in various settings, in the study of their linearized stability, and whether they survive under small perturbations of the problem. These singular solutions also appear as slow decay entire solutions of some equations such as the Lane-Edem-Fowler equation. I have worked on the existence of these slow decay solutions for this equation in exterior domains. There are related problems which have a similar structure, such as the ones obtained replacing the Laplacian by other operators. I have worked in identifying conditions for the existence and stability of singular solutions in these problems. The case of the Laplacian to the power one half can be viewed as a nonlinear Neumann equation, which appears with an exponential nonlinearity in geometry as well as in some applications to corrosion modeling. Another family of bifurcation questions is characterized by having a nonlinearity which is singular at a finite value such as 1/u. In the literature they turn up under the name of quenching problems, and recently they have attracted renovated interest in models of thin films and micro-electromechanical systems (MEMS). The singular nonlinearity is also relevant for the construction of symmetric minimal surfaces as observed by L. Simon. The role of singular solutions is played here by solutions that touch zero, and the questions are the existence and regularity of these solutions, linearized stability, bifurcation diagrams, and also the estimate of the size of the zero set, which in the thin film literature is interpreted as the rupture set.

Another line of research is related to reaction-diffusion equations and systems, which have been widely used to describe the dynamics of interacting populations. In the classical versions the spatial movement is modelled as a local phenomenon through differential operators. But in many situations in population ecology, such as in insect dispersal, the spread of individuals is better described as a long range process, which are better modeled by integral. Steady state and travelling wave solutions for single equations have been studied in the case of convolution operators with a symmetric kernel and some specific reaction nonlinearities. I am interested in the study logistic type equations for more general situations, characterizing the steady states and the asymptotic behavior of the solutions. The generalizations include nonsymmetric kernels, inhomogeneous equations, construction of pulsating waves, and models which combine nonlocal nonlinear interactions such as competition, self and cross diffusions.

[48] Hernán Castro, Juan Dávila, Hui Wang A Hardy Type inequality for $W^{m,1}_0(\Omega)$ functions J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 145–155. PDF

[47] J. Coville, J. Davila, S. Martinez Pulsating waves for nonlocal dispersion and KPP nonlinearity Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, 179–223. PDF

[46] Juan Davila, Dong Ye On finite Morse index solutions of two equations with negative exponent Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 121–128. PDF

[45] Juan Dávila, Luis López Regular solutions to a supercritical elliptic problem in exterior domains. J. Differential Equations 255 (2013), no. 4, 701–727.

[44] Juan Dávila, Manuel del Pino, Ignacio Guerra Non-uniqueness of positive ground states of non-linear Schrödinger equations Proc. Lond. Math. Soc. (3) 106 (2013), no. 2, 318–344.

[43] Resonance phenomenon for a Gelfand-type problem. Wenjing Chen, Juan D'avila Chen, Wenjing; Dávila, Juan  Nonlinear Anal. 89 (2013), 299–321.

[42] Juan Dávila, Juncheng Wei Point ruptures for a MEMS equation with fringing field Comm. Partial Differential Equations 37 (2012), no. 8, 1462–1493. PDF

[41] Juan Dávila, Marcelo Montenegro Concentration for an elliptic equation with singular nonlinearity J. Math. Pures Appl. (9) 97 (2012), no. 6, 545–578. PDF

[40] Chris Cosner, Juan Dávila, Salomé Martínez Evolutionary stability of ideal free nonlocal dispersal Journal of Biological Dynamics   PDF

[39] J. Davila, M. del Pino, M. Musso Bistable boundary reactions in two dimensions Arch. Ration. Mech. Anal. 200 (2011), no. 1, 89–140. PDF

[38] Juan Dávila, Louis Dupaigne, Alberto Farina Partial regularity of finite Morse index solutions to the Lane-Emden equation J. Funct. Anal. 261 (2011), no. 1, 218–232. PDF

[37] Antonio Capella, Juan Dávila, Louis Dupaigne, Yannick Sire Regularity of radial extremal solutions for some non local semilinear equations Comm. Partial Differential Equations 36 (2011), no. 8, 1353–1384. PDF

[36] J. Dávila, I. Peral Nonlinear elliptic problems with a singular weight on the boundary Calc. Var. Partial Differential Equations 41 (2011), no. 3-4, 567-586. PDF

[35] Juan Davila, Isabel Flores, Ignacio Guerra Multiplicity of solutions for a fourth order equation with power-type nonlinearity Math. Ann. 348 (2010), no. 1, 143--193. PDF

[34] J. Dávila, M. Montenegro Radial solutions of an elliptic equation with singular nonlinearity J. Math. Anal. Appl. 352 (2009), no. 1, 360--379 PDF

[33] Jerome Coville, Juan Davila Existence of radial stationary solutions for a system in combustion theory Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 3, 739-766. PDF

[32] Juan Dávila, Isabel Flores, Ignacio Guerra Multiplicity of solutions for a fourth order problem with exponential nonlinearit J. Differential Equations 247 (2009), no. 11, 3136--3162 PDF

[31] J. Dávila, L. Dupaigne, O. Goubet, S. Martínez Boundary blow-up solutions of cooperative systems Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, 1767--1791 PDF

[30] Jerome Coville, Juan Dávila, Salomé Martí­nez Existence and uniqueness of solutions to a non-local equation with monostable nonlinearity SIAM J. Math. Anal.  39  (2008),  no. 5, 1693--1709. PDF

[29] Jerome Coville, Juan Dávila, Salomé Martí­nez. Non-local anisotropic dispersal with monostable nonlinearity. J. Differential Equations 244 (2008), no. 12, 3080--3118. PDF

[28] Juan Dávila Singular solutions of semi-linear elliptic problems Handbook of Differential Equations Vol. 6 (2008), 83--176.

[27] Juan Dávila, Augusto C. Ponce. Hausdorff dimension of rupture sets and removable singularities. C. R. Math. Acad. Sci. Paris  346  (2008),  no. 1-2, 27--32. PDF

[26] Juan Dávila, Louis Dupaigne, Marcelo Montenegro The extremal solution of a boundary reaction problem Commun. Pure Appl. Anal. 7 (2008), no. 4, 795--817. PDF

[25] Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei Fast and slow decay solutions for supercritical elliptic problems in exterior domains. Calc. Var. Partial Differential Equations 32 (2008), no. 4, 453--480. PDF

[24] Juan Dávila, Michal Kowalczyk, Marcelo Montenegro Critical points of the regular part of the harmonic Green's function with Robin boundary condition J. Funct. Anal. 255 (2008), no. 5, 1057--1101. PDF

[23] Juan Dávila, Louis Dupaigne Perturbing singular solutions of the Gelfand problem Commun. Contemp. Math. 9 (2007), no. 5, 639--680. PDF

[22] Juan Dávila, Louis Dupaigne, Ignacio Guerra, Marcelo Montenegro Stable solutions for the bilaplacian with exponential nonlinearity SIAM J. Math. Anal.  39  (2007),  no. 2, 565--592.  PDF

[21] Juan Dávila, Manuel del Pino, Monica Musso The supercritical Lane-Emden-Fowler equation in exterior domains Communications in Partial Differential Equations, 32: 1225–1243, 2007  PDF

[20] Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei Standing waves for supercritical nonlinear Schrödinger equations Journal of Differential Equations 236 no. 1 (2007), 164-198. PDF

[19] Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei Singular limits of a two-dimensional boundary value problem arising in corrosion modelling Archive for Rational Mechanics and  Analysis 182 (2006) 181--221. PDF

[18] Juan Dávila, Manuel del Pino, Monica Musso Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data Journal of Functional Analysis 227 (2005), no. 2, 430--490 PDF

[17] Juan Dávila, Marcelo Montenegro Nonlinear problems with solutions exhibiting a free boundary on the boundary Ann. Inst. H. Poincaré Anal. Non Linéaire  22  (2005),  no. 3, 303--330. PDF

[16] Juan Dávila, Marcelo Montenegro Hölder estimates for solutions to a singular nonlinear Neumann problem Elliptic and parabolic problems,  189--205, Progr. Nonlinear Differential Equations Appl., 63, Birkhäuser, Basel, 2005. PDF

[15] Juan Dávila, Julián Fernández-Bonder, Julio D. Rossi, Pablo Groisman, Mariela Sued Numerical analysis of stochastic differential equations with explosions Stoch. Anal. Appl.  23  (2005),  no. 4, 809--825. PDF

[14] Juan Dávila, Marcelo Montenegro Existence and asymptotic behavior for a singular parabolic equation Trans. Amer. Math. Soc.  357  (2005),  no. 5, 1801--1828.

[13] Juan Dávila, Julio D. Rossi Self-similar solutions of the porous medium equation in a half-space with a nonlinear boundary condition: existence and symmetry J. Math. Anal. Appl.  296  (2004),  no. 2, 634--649.

[12] Juan Dávila Global regularity for a singular equation and local $H\sp 1$ minimizers of a nondifferentiable functional Commun. Contemp. Math.  6  (2004),  no. 1, 165--193. PDF

[11] Juan Dávila, Louis Dupaigne Hardy-type inequalities J. Eur. Math. Soc. (JEMS)  6  (2004),  no. 3, 335--365. PDF

[10] Juan Dávila, Augusto C. Ponce Variants of Kato´s inequality and removable singularities J. Anal. Math.  91  (2003), 143--178. PDF

[9] Juan Dávila, Marcelo Montenegro Positive versus free boundary solutions to a singular elliptic equation J. Anal. Math. 90 (2003), 303--335. PDF

[8] Juan Dávila, Louis Dupaigne Comparison results for PDEs with a singular potential Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 1, 61--83.

[7] Juan Dávila, Marcelo Montenegro A singular equation with positive and free boundary solutions RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.  97  (2003),  no. 1, 107--112.

[6] Juan Dávila, Radu Ignat Lifting of BV functions with values in $S\sp 1$ C. R. Math. Acad. Sci. Paris 337 (2003), no. 3, 159--164. PDF

[5] Juan Dávila On an open question about functions of bounded variation Calc. Var. Partial Differential Equations 15 (2002), no. 4, 519--527. PDF

[4] Juan Dávila A nonlinear elliptic equation with rapidly oscillating boundary conditions Asymptot. Anal. 28 (2001), no. 3-4, 279--307. PDF

[3] Juan Dávila A strong maximum principle for the Laplace equation with mixed boundary condition J. Funct. Anal. 183 (2001), no. 1, 231--244. PDF

[2] Juan Dávila Some extremal singular solutions of a nonlinear elliptic equation Differential Integral Equations 14 (2001), no. 3, 289--304. PDF

[1] Carlos Conca, Juan Dávila Optimal bounds for mixtures of infinitely many materials Numerical methods in mechanics (Concepción, 1995), 59--69, Pitman Res. Notes Math. Ser., 371, Longman, Harlow, 1997.