Fields of interest: I am interested in Nonlinear Elliptic Partial Differential Equations that appear in Mathematical Physics. In my work the main tools for this study are variational methods, topological methods, and perturbation arguments. For instance, I have worked on the Stationary Nonlinear
Schrödinger Equation (and other models alike) and the existence of semiclassical states. Moreover, I am involved in the application of variational methods to Liouville-type equations on compact surfaces that appear in Chern-Simons theory and Geometrical Analysis. More recently I am also working on semilinear elliptic problems under overdetermined boundary conditions.
Some slides: The following slides give a good idea of my recent work and my current interests:
- In 2011 I gave a conference at the University of Chile in Santiago.
- In 2013 I gave a conference at the "Congreso de Jóvenes Investigadores" held in Seville. This talk was adressed to a very heterogeneous audience, hence I tried to make it as accessible as possible.
- In 2015 I gave two talks at the RSME Conference held in Granada. One of them is this.
- In 2015 I gave a conference at the "Equadiff" held in Lyon (France).
- In 2017 I gave a conference at the 1st Belgium+Italy+Chile Conference in PDE’s held in Brussels (Belgium).
For any question, doubt or comment, please do not hesitate and contact me at
the address daruiz hat ugr.es.
- R. López-Soriano, A. Malchiodi and D. Ruiz, Conformal metrics with prescribed Gaussian and geodesic curvatures, preprint arXiv.org/abs/1806.11533.
- S. Cruz-Blázquez and D. Ruiz, "Prescribing Gaussian and geodesic curvature on the disk", preprint arXiv.org/abs/1806.06292, to appear in Advanced Nonlinear Studies.
- A. Ros, D. Ruiz and P. Sicbaldi, "Solutions to overdetermined elliptic problems in nontrivial exterior domains", preprint arXiv.org/abs/1609.03739,. to appear in J. European Math. Soc.
- D. Ruiz and J. Van Schaftingen, "Odd symmetry of least energy nodal solutions for the Choquard equation", J. Differential Equations 264 (2018), 1231-1262.
- F. De Marchis, R. López-Soriano and D. Ruiz, "Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials", J. Math. Pures Appliquées 115 (2018), 237-267.
- A. Ros, D. Ruiz and P. Sicbaldi, "A rigidity result for overdetermined elliptic problems in the plane" , Comm. Pure Appl. Math. 70 (2017), 1223-1252.
- T. D'Aprile, A. Pistoia and D. Ruiz, "Asymmetric blow-up for the SU(3) Toda System", J. Funct. Anal. 271 (2016), 495-531.
- Y. Jiang, A. Pomponio and D. Ruiz, "Standing waves for a gauged nonlinear Schrödinger equation with a vortex point", Comm. Contemp. Math 18 (2016), 1550074 (20 pages).
- R. López-Soriano and D. Ruiz, "Prescribing the Gaussian curvature in a subdomain of S^2 with Neumann boundary condition", J. Geom. Anal. 26 (2016), 630–644.
- T. D'Aprile, A. Pistoia and D. Ruiz, "A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up", Proc. London Math. Soc. 111 (2015), 797-830.
- L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz, "A general existence result for the Toda system on compact surfaces", Advances in Mathematics 285 (2015), 937-979.
- A. Malchiodi and D. Ruiz, "On the Leray-Schauder degree of the Toda system on compact surfaces", Proc. AMS. 143 (2015), 2985–2990.
- A. Pomponio and D. Ruiz, "Boundary concentration of a Gauged Nonlinear Schrodinger Equation", Calc. Var. PDE 53 (2015), 289-316
- A. Pomponio and D. Ruiz, "A Variational Analysis of a Gauged Nonlinear Schrödinger Equation", J. Eur. Math. Soc 17 (2015), 1463-1486.
- A. Malchiodi and D. Ruiz, “A variational Analysis of the Toda System on Compact Surfaces”, Comm. Pure Appl. Math. 66 (2013), 332-371.
- D. Ruiz, "A note on the uniformity of the constant in the Poincaré
inequality", special volume dedicated to A. Ambrosetti, Advanced Nonlinear Studies 12 (2012), 889-903.
- P. D'Avenia, A. Pomponio and D. Ruiz, “Semi-classical states for the Nonlinear Schrödinger Equation on saddle points of the potential via variational methods ”, Journal of Functional Analysis, 262 (2012), 4600-4633.
- I. Ianni y D. Ruiz, “Ground and bound states for a static Schrodinger-Poisson-Slater problem”, Comm. Contemp. Mathematics 14, (2012), No. 1.
- A. Malchiodi and D. Ruiz, “New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces”, GAFA 21 (2011), 1196-1217.
- T. D’Aprile and D. Ruiz, “Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems”, Mathematische Zeitschrift 268 (2011), 605-634.
- D. Ruiz y G. Vaira, “Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential”, Revista Matemática Iberoamericana 27 (2011), 253-271.
- D. Ruiz, “On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases”, Archive for Rational Mechanics and Analysis 198 (2010), 349-368.
- D. Ruiz and G. Siciliano, “Existence of ground states for a modified nonlinear Schrodinger equation”, Nonlinearity 23 (2010), 1221-1233.
- A. Ambrosetti and D. Ruiz , “Multiple bound states for the Schrödinger-Poisson problem”, Comm. Contemp. Math. 10 (2008), 391-404.
- A. Ambrosetti, G. Cerami and D. Ruiz, “Solitons of linearly coupled systems of semilinear non-autonomous equations on R^N” , Journal of Functional Analysis 254 (2008), 2816-2845.
- D. Ruiz and G. Siciliano, “A note on the Schrödinger-Poisson-Slater equation on bounded domains” , Advanced Nonlinear Studies 8 (2008), 179-190.
- D. Ruiz and A. Suárez, “Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term”, Proc. Royal Soc. Edinburgh Sect. A 137 (2007), 555-566.
- A. Ambrosetti, E. Colorado and D. Ruiz, “Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations”, Calculus of Variations and PDE 30 (2007), 85-112.
- D. Ruiz, “ The Schrödinger-Poisson equation under the effect of a nonlinear local term”, Journal of Functional Analysis 237 (2006), 655-674.
- A. Ambrosetti and D. Ruiz, “ Radial solutions concentrating on spheres of Nonlinear Schrödinger equations with vanishing potentials” , Proc. Royal Soc. Edinburgh 136 A (2006), 889-907.
- D. Arcoya and D. Ruiz, “The Ambrosetti-Prodi problem for the p-laplace operator”, Comm. in PDE 31 (2006), 849-865.
- A. Ambrosetti, A. Malchiodi and D. Ruiz, “ Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity”, Journal d'Analyse Mathématique 98 (2006), 317-348.
- A. Cañada and D. Ruiz, “ Asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance”, Journal of Math. Anal. and Appl. 313 (2006), 218-233.
- D. Ruiz, “ Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere”, Mathematical Models and Methods in Applied Sciences 15 (2005), 141-164.
- A. Ambrosetti, A. Malchiodi and D. Ruiz, "Recent trends on nonlinear elliptic equations on $R^n$" . Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 29 (2005), 3–13.
- A. Cañada and D. Ruiz, “ Periodic perturbations of a class of resonant problems”, Calculus of Variations and PDE 23 (2005), 281-300.
- D. Ruiz and J. R. Ward, “ Some notes on periodic systems with linear part at resonance”, Discrete and Continuous Dynamical Systems 11 (2004), 337-350.
- D. Ruiz, “ A priori estimates and existence of positive solutions for strongly nonlinear problems” , Journal of Diff. Eq. 199 (2004), 96-114.
- D. Ruiz, “ Resonant semilinear problems with nonlinear term depending on the derivative” , Journal of Math. Analysis and Appl. 295 (2004), 163-173.
- D. Bonheure, C. Fabry and D. Ruiz, “ Problems at resonance for equations with periodic nonlinearities ”, Nonlinear Analysis 55 (2003), no. 5, 557-581.
- D. Ruiz and M. Willem, “ Elliptic problems with critical exponent and Hardy potentials ”, Journal of Diff. Eq. 190 (2003), 524-538.
- J. Mawhin and D. Ruiz, “ A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction ”, Topol. Meth. in Nonl. Anal. 20 (2002), 1-14.
- A. Cañada and D. Ruiz, “ Resonant problems with multidimensional kernel and periodic nonlinearities ”, Diff. Int. Equations 16 (2003), 499-512.
- A. Cañada and D. Ruiz, “ Resonant nonlinear boundary value problems with almost periodic nonlinearity ”, B. Belgian. Math. Soc.-Simon Stevin 9 (2002), 193-204.