Martes 5 de Agosto 14:00hs, Aula de Seminarios, Pab I. Yannick  Sire (Universite Aix-Marseille 3, Paul Cezanne)

• Título: Symmetry of solutions for some non local equations and boundary reactions  
• Resumen: I will show the 1D symmetry (in the spirit of De Giorgi conjecture) of stable solutions for some non local equations involving the fractional laplacian in dimension 2. This is actually a by-product of a more general result concerning equations with reaction on the boundary, i.e. elliptic problems in the half-space with a nonlinear Neumann boundary condition.
  

  These are joint works with Enrico Valdinoci and Xavier Cabre

Martes 12 de Agosto 14:00hs, Aula de Seminarios, Pab I. Mayte Perez Llanos (Universidad Carlos III de Madrid)

• Título: Blow-up for a non-local diffusion problem with Newmann boundary conditions and a reaction term.
• Resumen: In this talk we study the blow-up problem for anon-local diffusion equation with a reaction term,
  $$
  

  u_t(x,t)=\int_\Omega J(x-y)(u(y,t)-u(x,t))\, dy+u^p(x,t).
  $$
  We prove that nonnegative and nontrivial solutions blow up in finite time if and only if $p>1$. Moreover, we find that the blow-up rate is the same that the one that holds for the ODE $u_t=u^p$, that is, $ \lim_{t \nearrow T} (T-t)^{\frac{1}{p-1}}\|u(\cdot, t)\|_{\infty} = (\frac{1}{p-1})^{\frac{1}{p-1}}$. Next, we deal with the blow-up set. We prove single point blow-up for radially symmetric solutions with a single maximum at the origin, as well as the localization of the blow-up set near any prescribed point, for certain initial conditions in a general domain with $p>2$. Finally, we show some numerical experiments which illustrate our results.
  Este es un trabajo conjunto con Julio D. Rossi.

• F.~Andreu, J.~M. Maz\'{o}n, J.~D. Rossi and J.~Toledo. {\it The Neumann problem for nonlocal nonlinear diffusion equations.} To appear in J. Evol. Equations.
• F.~Andreu, J.~M. Maz\'{o}n, J.~D. Rossi and J.~Toledo. {\it A nonlocal $p-$Laplacian evolution equation with
  Neumann boundary conditions}. To appear in J. Math. Pures Appl
•  P. Bates and A. Chmaj. {\it An integrodifferential model for phase transitions: stationary solutions in higher
  dimensions}. J. Statistical Phys., {\bf 95}, (1999), 1119--1139.
•  P. Bates, P. Fife, X. Ren and X. Wang. {\it Travelling waves in a convolution model for phase transitions}. Arch. Rat. Mech. Anal., {\bf 138}, (1997), 105--136.
•  C. Bandle and H. Brunner. {\it Blow-up in diffusion equations: a survey.} J. Comp. Appl. Math., {\bf 97}, (1998), 3--22.
•  C. Carrillo and P. Fife. {\it Spatial effects in discrete generation population models}. J. Math. Biol., {\bf 50(2)}, (2005), 161--188.
•  E. Chasseigne, M. Chaves and J. D. Rossi. {\it Asymptotic behaviour for nonlocal diffusion equations.} J. Math. Pures Appl., {\bf 86}, (2006), 271--291.
•  X. Chen. {\it Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations}.
  

  Adv. Differential Equations, {\bf 2}, (1997), 125--160.

•  X. Y. Chen and H. Matano. {\it Convergence, asymptotic periodicity and finite point blow up in one-dimensional semilinear heat equations}. J. Differential Equations, Vol. 78, (1989), 160--190.
•  C. Cortazar, M. Elgueta and J. D. Rossi. {\it A non-local diffusion equation whose solutions develop a free boundary.} Annales Henri Poincar\'{e}, {\bf 6(2)}, (2005), 269--281.
•  C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski. {\it Boundary fluxes for non-local diffusion.} J. Differential
  Equations, {\bf 234}, (2007), 360--390.
•  C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski. {\it How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems.} To appear in Arch. Rat. Mech. Anal.
• P. Fife. {\it Some nonclassical trends in parabolic and parabolic-like evolutions}. Trends in nonlinear analysis,
  153--191, Springer, Berlin, 2003.
• P. Fife and X. Wang. {\it A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions}. Adv. Differential Equations, {\bf 3(1)}, (1998), 85--110.
•  V. Galaktionov and J.~L.~V\'{a}zquez. {\it The problem of blow-up in nonlinear parabolic equations}, Discrete Contin. Dynam. Systems A, {\bf 8}, (2002), 399--433.
•  P. Groisman and J. D. Rossi. {\it Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions.} J. Comput. Appl. Math. {\bf 135}, (2001), 135--155.
• Friedman, A. y McLeod, B. {\it Blow-up of positive solutions of semilinear heat equations.} Indiana Univ. Math. J. {\bf 34}, no. 2 (1985), 425--447.
• L. I. Ignat and J. D. Rossi. {\it A nonlocal convection-diffusion equation}. J. Functional Analysis, {\bf 251(2)}, (2007), 399--437.
• A. Samarski, V. A. Galaktionov, S. P. Kurdyunov and A. P. Mikailov. Blow-up in quasilinear parabolic equations. Walter de Gruyter, Berlin, (1995).

Miercoles 23 de Julio 14:00hs, Aula de Seminarios, Pab I. Luis Silvestre (Courant Institute of Mathematical Sciences)

• Título:  Ecuaciones íntegro-differenciales no lineales.
• Resumen: En esta charla vamos a ver algunos resultados de regularidad para ecuaciones íntegro-differenciales. Estas ecuaciones aparecen naturalmente cuando uno estudia problemas de control con procesos estocasticos discontinuous. El caso mas común es el laplaciano fraccionario, pero nosotros nos vamos a concentrar en problemas no lineales. Estas ecuaciones se comportan de manera muy similar a las EDPs elípticas, y la gran diferencia es que no son locales. Vamos a obtener en este contexto teoremas análogos a la desigualdad de Alexandroff, la desigualdad de Harnack de Krylov y Safonov, y estimaciones C^{1,alfa}. Las ecuaciones en derivadas parciales se pueden obtener como casos límites de ecuaciones íntegro-differenciales, por lo que nuestros resultados son una generalización de la teoría de regularidad para ecuaciones elípticas no linales al caso no local. Es un trabajo en conjunto con Luis Caffarelli.

Martes 1de Julio 14:00hs, Aula E24, Pab I. Richard Moore (New Jersey Inst. of Tech)

• Título: Finding Failures in Optical Fibre Lines.
• Resumen: Prior to installation, fibre-optic communication lines must be tested for robustness against environmental sources of noise; typically, industry standards require that no more than one bit in every billion be lost in transmission. Even where good physical models for transmission exist, they typically involve stochastic nonlinear partial differential equations that render typical simplifying assumptions (e.g., that the distributions are Gaussian) on transmitted pulse parameters (energy, position, etc.) invalid. Performing simple Monte Carlo simulations on these systems is equally problematic due to the computational expense involved.
  

  We present a method that uses finite-dimensional reductions based on perturbation theory and variational techniques in combination with straightforward sampling methods to build a hybrid analytical-computational approach to resolving these statistically rare events in optical communication lines. Depending on the quality of the finite-dimensional reduction, the efficiency of this method is several orders of magnitude greater than simple Monte Carlo simulations for a given accuracy. This is joint work with Gino Biondini (SUNY Buffalo) and Bill Kath (Northwestern Univ.).

Martes 3 de Junio 14:00hs, Aula E24, Pab I. Joana Terra (Universidad de Buenos Aires)

• Título: Estabilidad de soluciones tipo silla y soluciones minimales.
• Resumen: Las soluciones tipo silla aparecen como un posible contra-ejemplo a una conjectura de De Giorgi. Consideramos una familia de ecuaciones elipticas semilineales. Definiremos solucion tipo silla en todas dimensiones pares y mostraremos su existencia, monotonia y comportamiento asintotico. Ademas, probamos que tales soluciones son inestables en dimensiones 4 y 6. En dimension 2 el resultado ya era conocido, aunque con un metodo distinto. En el caso de problemas no variacionales tambien se puede definir estabilidad de soluciones. Estudiaremos existencia y regularidad de soluciones estables minimales de una ecuacion no variacional con un termino cuadratico en el gradiente

Miércoles 14 de Mayo, 14:00hs, Aula E24, Pab I. Malena Español (Tufts University, Medford)

• Título: Metodo multilevel para problemas discretos ill-posed.
• Resumen:Los problemas discretos ill-posed en la forma de sistemas lineales o de minimos cuadrados ocurren en diversas aplicaciones, por ejemplo en la reconstruccion de señales o imagenes. La dificultad en resolver este tipo de problemas se debe a la presencia de ruido. En esta charla vamos a discutir las propiedades de estos problemas y estudiar metodos para aproximar soluciones regularizadas. En particular, presentaremos un metodo multilevel para aproximar soluciones regularizadas para el problema de reconstruccion de se~nales para operadores con estructura Toeplitz. Este metodo se basa en los metodos multigrilla estandares desarrollados para resolver ecuaciones diferenciales y hace uso de la transformada Haar wavelet para moverse entre grillas. El uso de la transformada Haar wavelet permite mantener la estructura Toeplitz en todos los niveles lo cual hace a este metodo computacionalmente competitivo.

Viernes 2 de Mayo, 14:30hs, Aula de Seminarios, Departamento de Matemáticas: Elvira Mascolo (Full Professor in Mathematical Analysis Dept. of Mathematics "Ulisse Dini", University of Firenze, Italy)

• Título: A short userguide for the study of polyconvex functionals.
• Resumen: Direct methods of Calculus of Variations are a rigorous framework to treat mathematically interesting variational problems arising in science or engineering. The methods are based on the notion of lower semicontinuity, which classically is based on the convexity of the integrad function however this assumptions is unrealistic in the framework of nonlinear elasticity. Then it must be replaced by different and more general conditions, named quasiconvexity and polyconvexity. In this conference I introduce some new recent semicontinuity results for non coercive polyconvex functionals. Moreover we present some existence results for polyconvex and also non polyconvex problems, which are related with the Odgen model for Rubberlike Materials. This existence results are strictly connected with the resolution of the Prescribed Volume Equation.

Lunes 21 de Abril, 17hs, Aula de Seminarios, Departamento de Matemáticas:

Humberto Ramos Quoirin (Universitè Libre de Bruxelles )

• Título: A weighted eigenvalue problem for an indefinite elliptic operator.
• Resumen:Let $\Delta_p u=div(|\nabla u|^{p-2}\nabla u)$ be the p-Laplacian operator and $\Omega$ be a bounded domain in $R^N$. Given a potential V and a weight m, the eigenvalue problem
  (P) $\Delta_pu+V(x) |u|^{p-2} u=\lambda m(x) |u|^{p-2}u,\quad u\in W_0^{1,p}(\Omega)$
  may not have a principal eigenvale. We find a necessary and sufficient condition on V and m for the existence of principal eigenvalues of (P). Further eigenvalues are also investigated.

Jueves 13 de Marzo, 16hs, Aula de Seminarios: José Miguel Urbano (CMUC/University of Coimbra, Portugal)

• Título: Intrinsic scaling for nonlinear pdes arising in the modeling of chemotaxis, immiscible fluids and phase transitions.
• Resumen:Three different problems, arising from chemotaxis, the flow of immiscible fluids and phase transitions, are presented. The problems correspond to a wide range of applications and share a common feature: they are modeled by (systems) of nonlinear partial differential equations with some sort of degeneracy or singularity (in some cases, both). A survey of methods tailored to balance the structure of the equations and the intrinsic geometric configuration in which they must be analyzed is described. The goal is to obtain regularity estimates that play a crucial role in the understanding of all the phenomena under study.