RISM (Riemann International School of Mathematics)
organizes the meeting
Verbania (Italy) September 26 - October 1, 2010


Lectures

Alice Chang (Princeton University)
On Alexendrov-Fenchel inequality for k-convex domain
Abstract: pdf format (9 KB)

Djairo de Figueiredo (Univ. de Campinas, Sao Paulo)
Biharmonic equation with nonlinearities of the Henon type
Abstract: We discuss existence, multiplicity and regularity of solutions for the biharmonic under both Dirichet and Navier boundary conditions.
The techniques area variational. So in the case of radial solutions we prove some embeddings of weighted Sobolev spaces of radial functions. Multiplicity is obtained by the use of Sobolev spaces with partial symmetries.

Manuel del Pino (Univ. de Chile)
New entire solutions of semilinear elliptic equations
Abstract: We discuss some new results on construction of families solutions with interesting asymptotic patterns for some problems including the Allen-Cahn and the standing-wave focusing NLS equations. In particular the correspondence of families of solutions with classes of complete embedded minimal surfaces will be described.

Yan yan Li (Rutgers Univ.)
Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities
Abstract: We present some results on the behavior of positive solutions in a punctured ball of general second order fully nonlinear conformally invariant elliptic equations. We prove that such a solution, near the puncture, is asymptotic to some radial solution of the same equation in the punctured Euclidean space.
This is a joint work with Z.C. Han and E. Teixeira.

Chang-Shou Lin (National Taiwan Univ)
A Topological Degree Counting Formulas for a Liouville System
Abstract: We consider a system of nonlinear pde equations in two dimensional space. The nonlinear term involves the exponential functions. For a single equation, it is the classical Liouville equation. For this system, we will the aprior bounds for all solutions when the parameters are non-critical or equivalently, we should determine the set of non-critical parameters. We also derive a topological degree counting formulas for this Liouville systems. As the consequence, we can prove the existence of solutions when the Euler chacteristic of the domain is non-positive. This is a joint work with Lei Zhang.

Stefan Hildebrandt (Universität Bonn)
On Plateau's Problem and Riemann's Mapping Theorem

Jean Mawhin (Université Catholique de Louvain)
Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities
Abstract: pdf format (24 KB)

Louis Nirenberg (CIMS, New York)
Remarks on singular solutions of fully nonlinear elliptic and parabolic operators
Abstract: Various forms of the maximum principle, including the strong maximum principle and the Hopf Lemma are established for some singular solutions ..Some applications are given to proving monotonicity and symmetry of singular solutions using the method of moving planes.

Paul Rabinowitz (Univ. of Wisconsin, Madison)
Global and local minimizers for a class of variational problems
Abstract: We discuss the use of methods from the calculus of variations, PDE, dynamical systems, and geometry to construct and study solutions for a class of PDE's introduced by Moser. The existence and multiplicity of period, heteroclinic and homoclinic solutions as global and local minimizers of corresponding variational problems will be explored.

Jalal Shatah (CIMS, New York)
Space Time Resonance Method and Water Waves Problem
Abstract: We introduce the concept of space time resonant sets for evolution equations and develop a method to prove global existence for nonlinear problems based on space time resonance calculations.
We show how the space time resonance method can be used to explain and prove many of the known results on global existence of small solutions to nonlinear dispersive and hyperbolic equations.
We also show how this method is used to prove that small amplitude surface waves exist globally. We include a full discussion of the fluid interface problems.

Michael Struwe (ETH, Zurich)
A "supercritical" nonlinear wave equation in 2 space dimensions
Abstract: Extending the work of Ibrahim et al. on the Cauchy problem for wave equations with exponential nonlinearities in 2 space dimensions, we show how global well-posedness may be established also in the "supercritical' regime of large energies.

Giorgio Talenti (Univ. di Firenze)
Remarks on Busemann equation
Abstract: pdf format (53 KB)

Paul Yang (Princeton University)
A fourth order invariant in CR geometry in 3-D
Abstract: In CR geometry in 3-D, there is strong analogy with conformal geometry of dimension four. In particular, there are two conformally covariant operators which play critical role in the analysis: the analogue of the conformal Laplacian as well as the analogue of the fourth order operator of Paneitz. In recent work we found the positivity of both these operators have strong consequences: the positivity of mass as well as the imbeddability of CR structures.



Seminars

Claudianor O. Alves (Universidade Federal de Campina Grande)
Abstract: pdf format (14 KB)

Elves A. B. Silva (Departamento de Matemática, Universidade de Brasília)
Existence of a positive solution for quasilinear Schrödinger equations
Abstract: pdf format (32 KB)

Vieri Benci (Universitŕ di Pisa)
Hylomorphic solitons for the Klein-Gordon-Maxwell equations
Abstract: Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. The solitons whose existence is related to the ratio energy/charge are called hylomorphic. This class includes the Q- balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation, as well as the solitons which occur, by the same mechanism, in the nonlinear Schroedinger equation. We will show that also the nonlinear Klein- Gordon-Maxwell equations admits hylomorphic solitons. This kind of soliton in the physical literature are called charged Q-balls.

Joao Marcos Bezerra do Ó (Universidade Federal da Paraíba)
A Sharp Trudinger-Moser Type Inequality
Abstract: pdf format (13 KB)

Lucio Boccardo (Universitŕ di Roma 1)
Quasilinear and semilinear singular elliptic equations and systems
Abstract: pdf format (91 KB)

Alfonso Castro (Harvey Mudd College, Claremont)
Wave equations with non-monotone nonlinearity
Abstract: Recent results on existence of solutions to wave equations with non-monotone nonlinearity and infinite dimensional kernel will be discussed. Loss of regularity due to the interaction of the nonlinearity with eigenvalues of infinite multiplicity will be presented.

Giovanna Cerami (Politecnico di Bari)
Multiple positive solutions for some Schröodinger Equations
Abstract: pdf format (20 KB)

Mónica Clapp (Universidad Nacional Autónoma de México)
Intertwining semiclassical solutions to a Schrödinger-Newton system
Abstract: pdf format (45 KB)

David Costa (University of Nevada)
On Positive Solutions for a Class of Caffarelli-Kohn-Nirenberg type Equations
Abstract: We consider solvability for a class of singular Caffarelli-Kohn-Nirenberg type equations in R^N with a sign-changing weight function. In particular, we examine how the properties of the Nehari manifold and the fibrering maps affect the question of existence of positive solutions.

Arnaldo S. do Nascimento (Universidade Federal do Sao Carlos)
Stable stationary solutions to a reaction-diffusion equation with zero Neumann boundary condition in rectangular domains.
Abstract: We prove existence of nonconstant stable stationary solutions of a reaction-diffusion equation with no flux boundary condition in rectangles and squares. By a well-known result of Casten and Holland and also Matano, such solutions do not exist in convex domains which are at least C^2; we prove the same result under weaker hypothesis on the regularity of the domain. In particular we exhibit a C^1 convex domain for which this result still holds.
The approach is variational and based on Gamma-convergence techniques. Symmetry inheritance from symmetry properties of the domain and the reaction term is also obtained using the Unique Continuation Principle. In particular for the square a complete geometric picture of the level sets of the stable equilibria is obtained.

Marcelo F. Furtado (Universidade de Brasília)
Multiplicity and concentration of solutions for elliptic systems with vanishing potentials
Abstract: pdf format (31 KB)

Filippo Gazzola (Dipartimento di Matematica Politecnico di Milano)
The first biharmonic Steklov eigenvalue: optimization and critical growth elliptic problems
Abstract: We study some properties of the first biharmonic Steklov eigenvalue.
We set up an optimal shape problem and we show its role in critical growth semilinear elliptic problems.

Jean-Pierre Gossez (Univ. of Bruxelles)
Maximum and antimaximum principles: beyond the frst eigenvalue
Abstract: pdf format (31 KB)

Norimichi Hirano (Yokohama National University)
Existence of Steady Stable Solutions for Ginzburg-Landau equation in a domain with nontrivial topology
Abstract: pdf format (37 KB)

Otared Kavian (Universite de Versailles)
A Nonlinear Population Dynamics Model: Approximate Controllability by Birth Control using the Kakutani Fixed Point Theorem
Abstract: we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals.
In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem for multivalued operators.

Sebastián Lorca (UTA, Chile)
Multiple solutions for the mean curvature equation
Abstract: pdf format (14 KB)

Liliane Maia (Departamento de Matemática, Universidade de Brasília, Brazil)
Existence of antisymmetric solutions for a class of nonlinear Schrödinger equations
Abstract: pdf format (44 KB)

Gianni Mancini (Universitŕ degli Studi ”Roma Tre”)
Moser-Trudinger inequalities on conformal discs
Abstract: pdf format (65 KB)

Everaldo Souto de Medeiros (Universidade Federal da Paraěba)
Weak solutions of quasilinear elliptic systems via the cohomological index
Abstract: pdf format (26 KB)

Anna Maria Micheletti (Univ. di Pisa)
Some generic properties of non degeneracy of critical points
Abstract: pdf format (48 KB)

Olimpio Miyagaki (Universidade Federal de Juiz de Fora)
Existence results for quasilinear elliptic exterior problems involving convection term with nonlinear Robin boundary conditions
Abstract: pdf format (45 KB)

Marcelo Montenegro (Unicamp, Brazil)
Concentrating solutions for an elliptic equation arising in free boundary problems
Abstract: pdf format (13 KB)

Filomena Pacella (Univ. Roma 1)
Radial solutions of semilinear elliptic problems: spectral analysis and bifurcation
Abstract: We consider radial positive solutions of semilinear elliptic problems with power nonlinearities either in an annulus or in the exterior of a ball and show some results on the spectrum of the linearized operator. From this we deduce on one side bifurcation results, in particular with respect to the exponent of the nonlinear term, on the other side existence of "quasiradial solutions" in domains close to an annulus. All results apply also to the supercritical case when the existence and multiplicity of (nonradial) solutions seems particularly difficult to get.

Charles Stuart (Ecole Polytechnique Fédérale Lausanne)
Localization of P-S and Cerami sequences in a mountain pass geometry
Abstract: pdf format (34 KB)

Gabriella Tarantello (Universitŕ di Roma Tor Vergata)
Uniqueness and Symmetry results for solutions of a mean field equation on s^2 via a new bubbling phenomenon
Abstract: Motivated by the study of gauge field vortices we consider a mean field equation on the standard two-sphere involving a Dirac distribution supported at a point P of S^2.
As needed for the applications, we show that solutions ”concentrate” exactly at P for some limiting value of a given parameter. We use this fact to obtain symmetry (about the axis OP) and uniqueness property for the solution. The presence of the Dirac measure makes such a task particularly delicate. Indeed, we need to rule out the possibility that, after blow up (in a suitable scale), the solution sequence may admit a double ”peak” profile described in terms of appropriate “limiting” problems. For instance we need to account for the presence of non-axially symmetric solutions in such ”limiting” problems. In this process, we establish a symmetry result about a maximal circle through P and its antipodal point P*, that applies to more general situations where the full axial symmetry cannot be expected.
This work is in collaboration with D. BARTOLUCCI and C.S. LIN.

Eduardo Teixeira (Univ. Federal Ceara)
Existence and regularity theory for free boundary problems in Riemannian manifolds
Abstract: Physical models in Riemannian manifolds are central themes of research within the modern theory of mathematical analysis. Problems with unknown boundaries (free boundary problems), often employed to model discontinuous change of phases, are of particular interest in this new set up. In this talk, I will present some fine regularity tools, developed together with Lei Zhang (Univ. Florida), in the study of free boundary problems in non-Euclidean environments. Our results are generalizations of the famous almost-monotonicity formulae of Caffarelli, Jerison and Kenig (Ann Math 2002). Several applications will be delivered.

Susanna Terracini (Univ. Milano Bicocca)
On some optimal partition problems
Abstract: We consider the free boundary problem associated with optimal partitions related with linear and nonlinear eigenvalues. We are concerned with extremality conditions and the regularity of the nodal sets, also in connection with that of eigenfunctions and the number of their nodal domains.

Pedro Ubilla (Universidad de Santiago de Chile)
Superlinear elliptic problems with sign changing coefficients
Abstract: pdf format (20 KB)