Schedule

 Héctor Chang (CIMAT)

 Title: Regularity Estimates for Zeroth Order Operators.

 Carlos Culebro (UNAM)

Title: Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity

 Briceyda Delgado (INFOTEC)

 Title: Maximal and minimal weak solutions for certain elliptic problems

 Abstract: We consider a semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. Moreover, we will explore the possibility of applying this method for other equations of Mathematical Physics.

 Jorge Faya  (Universidad Austral de Chile)

 Title: Critical Equations with a Sharp Sign Change in the Nonlinearity

Juan Carlos Fernández (FC - UNAM)

Title: Schwarz symmetrization in cohomogeneity one manifolds

Abstract: The symmetrization techniques for domains and functions have proven useful in several areas, including Geometric Analysis. In this talk we extend the usual Schwarz symmetrization in Euclidean spaces to closed riemannian manifolds admitting a cohomogeneity one isometric action of a Lie group, that is, every principal orbit is a hypersurface with constant mean curvature of the manifold. With this machinery, we discuss suitable versions of some classical geometric inequalities such as the isoperimetric, Pólya-Szegö and Hardy-Littlewood inequalities, as well as a version of the Talenti comparison theorem. 

Aria Halavati (Courant Institute of Mathematical Sciences)

 Title: Decay of Excess for the Abelian Higgs Model

 Abstract: Entire critical points of the Abelian-Higgs functional are known to blow down to (generalized) minimal submanifolds of codimension 2. We focus on the multiplicity-one regime, where the blow-down consists of a single sheet. In this talk, I will present results from three papers that together provide the analytical tools to prove that entire critical points in this regime exhibit an improvement of flatness, leading to the uniqueness of blow-downs.

We further show that stationary points are flat in low dimensions, using an Allard-type argument, while local minimizers are flat in all dimensions via a De Giorgi-type method. This can be viewed as a large-scale regularity theorem in the spirit of geometric measure theory.

As part of this analytical framework, I will also discuss new weighted inequalities on two-manifolds, as well as a quantitative stability theorem for the Abelian-Higgs model in dimension 2. This result is not only similar in spirit to the quantitative isoperimetric inequality, but also inspires from it in technique.

 Víctor Hernández-Santamaría (IMUNAM - CU)

 Title: Some unique continuation properties for weakly coupled elliptic systems

 Abstract:  In this talk, we will discuss the strong unique continuation property for weakly coupled elliptic systems, with an emphasis on competitive systems. Using Carleman estimates, we show that if a solution to such a system vanishes on an open subset, it must be trivial on the whole domain. I will briefly review the concept of Carleman estimates and explain how the system's structure is exploited in our proofs. This approach opens the door to prove some nonexistence results, highlighting the impact of continuation properties on elliptic systems with critical nonlinearities.

Luis Fernando López Ríos ( IIMAS - UNAM)

Title: Stability of traveling waves for Allen-Cahn equation with phase-dependent diffusivity 

Abstract: In this talk, I will present one-dimensional Allen-Cahn equations with phase-dependent diffusivity (possibly degenerate) and analyze the stability of their traveling waves. As in the classical Allen-Cahn equation (also known in different contexts as Nagumo or Ginzburg-Landau), these types of solutions are among the most important and studied.

We will consider first the dynamics in the whole space, where the traveling waves are free to move, and establish their spectral stability. Then we focus our attention on bounded intervals, where the waves hit the boundary and exhibit layer transitions that maintain their structure for a very long time to eventually collapse to a pure constant solution, one of the two possible stable equilibriums of the Allen-Cahn equation, which are also minimizers of the Ginzburg-Landau energy functional. This phenomenon is known as metastability.

 Liliane Maia (Universidade de Brasília)

 Title: Advances in solving nonlinear Schrödinger equations with general potentials

 Sean Ries McCurdy (IM - UNAM)

Title: Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index

 Abstract:  Questions about minimal surfaces have been central to the development of non-linear PDEs and Geometric Measure Theory.   In the last several decades, much interest has centered on generalizing the ideas developed for minimal surfaces to the context of "almost-minimal" surfaces.  One way to formulate "almost-minimality" is through the variational notion of Bounded Index.  This talk will present an introduction to some recent results (joint with Nicolau Aiex and Paul Minter) extending the estimates of A. Naber and D. Valotrta on the singular set of minimal hypersurfaces (i.e., minimizing integral currents) to the case of integral varifolds of bounded index.  These results rely upon the powerful regularity theory developed by N. Wickramasekera.  The emphasis of the talk will be on presenting the background and big ideas.  There will be more pictures than equations.

 Cristian Morales-Encinos (IM UNAM)

 Title: On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity 

 Isidro Munive (Universidad de Guadalajara)

 Title: Multiplicity of 2-nodal solutions of the Yamabe equation

 Abstract:  In this talk we prove multiplicity results for 2-nodal solutions of a subcritical non-linear equation on a closed Riemannian manifold (M,g). If (N,h) is a closed Riemannian manifold of constant positive scalar curvature our result gives multiplicity results for the Yamabe-type equation on the Riemannian product (M × N, g + εh), for ε > 0 small.

Gabrielle Nornberg (Universidad de Chile)

 Title: Some unique continuation results for nonlinear problems

 Abstract: In this talk we discuss some unique continuation results which appear in the context of nonexistence of solutions for nonlinear elliptic problems.

 Cintia Pacchiano (IM - UNAM Cuernava)

 Title: Regularity Results for Double Phase Problems on Metric Measure Spaces

 Abstract: In this talk, we present boundedness, Hölder continuity and Harnack inequality results for local quasiminima to elliptic double phase problems of p-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach, based on the De Giorgi method and a careful phase analysis. The main novelty is the use of an intrinsic approach, based on a double phase Sobolev-Poincaré inequality.

Furthermore, we present boundary regularity results for quasiminimizers of double-phase functionals.  We again use a variational approach to give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. This in an on-going project, together with Prof. Dr. Antonella Nastasi from University of Palermo.

During the past two decades, a theory of Sobolev functions and first degree calculus has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc. 

Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to disciplines as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.

Kanishka Perera (Florida Institute of Technology)

Title:  Variational methods for scaled problems with applications to the Schrodinger–Poisson–Slater equation

 Jimmy Petean (CIMAT)

 Title: Global Bifurcation for the Paneitz-Branson equation

 Abstract: The Q-curvature and the associated Paneitz-Branson equation on Riemannian manifolds appeared in the study  of conformally invariant operators. They are seen as fourth order equivalents of the scalar curvature and the associated Yamabe equation. It is interesting to understand if techniques used in the case of the Yamabe equation can be applied in the fourth order case. In both cases there are interesting trivial families of solutions on certain manifolds and it is natural to consider bifurcation from these families. In the talk I will present results obtained with Jurgen Julio Batalla on global bifurcation for the Paneitz-Branson equation, which requires a qualitative understanding of the families of solutions of a fourth order ODE bifurcating from the trivial family.

 Mayra Soares (Universidade de Brasília)

Title: A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior 

Víctor Alfonso Vicente Benítez (IM - UNAM Juriquilla)

Title: Entire solutions to a quasilinear purely critical competitive system 

Abstract: In this talk, we present recent results on the existence and nonexistence of fully nontrivial least energy solutions for a quasilinear, purely critical competitive system involving the p-Laplacian.  We establish the existence of pinwheel solutions, that is, fully nontrivial solutions that are invariant under a certain group of isometries and for which each component can be obtained from the first one via a linear isometry. Finally, we show that a quasilinear equation associated with the p-Laplacian admits an infinite many nodal solutions. This talk is based on joint work [1] with Mónica Clapp (IM - UNAM  Juriquilla)