
Ohio State University Partial Differential Equations
Seminar
Year 20132014
Time/Location: Wednesdays 4:105:05pm/MW154
(unless otherwise noted) 





TIME 
SPEAKER 
TITLE 
HOST 
August 21 
No seminar 


August 28 
No seminar 


September 4 
No seminar 


September 11 
No Seminar 

September 18 
No Seminar 

September 25 
No Seminar 

October 2 
Chris Cosner

Evolutionary Stability of Ideal Free Dispersal Strategies: A Nonlocal
Dispersal Model 
Yuan
Lou 
October 9 
Steve Cantrell 
Avoidance Behavior in Intraguild Predation Commurnities: A Cross Diffusion Model 
Yuan
Lou 
October 16 
No Seminar 


October 23 
No Seminar 


October 30 
No Seminar 


November 6 
Marco Fontelos 
Mathematical modelling of Electrowetting phenomena  Avner Friedman 
November 13 
Mariana Smit Vega Garcia 
New developments in the thin obstacle problem with Lipschitz coefficients  Barbara Keyfitz 
November 20 
Bo Guan 
The concavity and subsolution in estimates for fully nonlinear elliptic equations  
November 27 
Thanksgiving 

December 4 
No seminar 


January 8 
No Seminar 


January 15 
No seminar 


January 22 
Bo Guan 
The Dirichlet Problem for Fully Nonlinear Elliptic Equations 

January 29 
No Seminar 


February 5 
Bo Guan 
The Dirichlet Problem for Fully Nonlinear Elliptic Equations II 

February 12 
Feride Tiglay 
Integrable Evolution Equations on Spaces of Tensor Densities 
Barbara Keyfitz 
February 19 
Avner Friedman 
The Dynamics of Ant Trails 

February 26 
Stephen Preston 
Higherdimensional HunterSaxton and CamassaHolm equations 
Feride Tiglay 
March 5 
No Seminar 


March 12 
No Seminar 
Spring Break  
March 19 
Xiangwen Zhang 
A Proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R^3  Bo Guan 
March 20 
Yaping Wu 
Steady States and Traveling Waves for SKT Competition Model with CrossDiffusion  Yuan Lou/Adrian Lam 
March 26 
No Seminar 

April 2 
Marcus Khuri 
The massangular momentum inequality for asymptotically flat and asymptotically hyperbolic initial data  Bo Guan 
April 9 
Richard Wentworth 
The YangMills flow on Kaehler manifolds  Bo Guan 
April 16 
Lu Xu 
BrunnMinkowski inequalities and related problems 
Bo Guan 
April 17 
Yong Huang 
The Lp Minkowski problem 
Bo Guan 
April 23 
Wei Zhou 
On the regularity for the Dirichlet problem for degenerate Hessian equations  Bo Guan/Adrian Lam 
Chris Cosner (University of Miami) on October 2, 2013
Title: Evolutionary Stability of Ideal Free Dispersal Strategies: A Nonlocal
Dispersal Model
Abstract: The dispersal of organisms has many significant ecological effects, and
hence the evolution of dispersal has been a subject of considerable
interest in evolutionary ecology. An important problem in the study of the
evolution of dispersal is determining what kinds of dispersal strategies
are evolutionarily stable in the sense that populations using them cannot
be invaded by ecologically similar populations using other strategies. A
class of strategies that have been shown to be evolutionarily stable in
various contexts are those that produce an ideal free distribution of the
population, that is, a spatial distribution where no individual can
increase its fitness by moving to another location. This talk will present
results on the evolutionary stability of ideal free dispersal strategies
in the context of continuous time nonlocal dispersal models. These results
partially extend some recent work on the evolutionary stability of ideal
free dispersal for reactionadvectiondiffusion equations and discrete
diffusion models to nonlocal dispersal models. They also include an
extension of an inequality from matrix theory to the case of nonlocal
dispersal operators, which may be of independent interest.
Steve Cantrell (University of Miami) on October
9, 2013
Title: Avoidance Behavior in Intraguild Predation Commurnities: A Cross Diffusion Model
Abstract: A crossdiffusion model of an intraguild predation community, in which the intraguild prey species employs a fitness based avoidance strategy, is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor are proved. It is shown that if the intraguild prey has positive fitness at any point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states where the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems. This is joint work with Dan Ryan at NIMBIOS, University of Tennessee.
Marco Fontelos (Universidad Autonoma de Madrid) on November 6, 2013
Title: Mathematical modelling of Electrowetting phenomena
Abstract: The term electrowetting is commonly used for some techniques to change the shape and wetting behaviour of liquid droplets by the application of electric fields and charges. First, we describe the presence of symmetry breaking bifurcations and their physical role. We then develop and analyze a model for electrowetting that combines the Navierâ€“Stokes system for fluid flow, a phasefield model of Cahnâ€“Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. A critical role in the deduction of suitable couplings between phase field and other physical fields is played by variational principles similar to those that apply for gradient flows. A consequence of such principle is the deduction of energy estimates that serve to prove existence and uniqueness of solutions for the resulting system of equations.
Mariana Smit Vega Garcia (Purdue University) on November 13, 2013
Title: New developments in the thin obstacle problem with Lipschitz coefficients
Abstract: We will describe the lowerdimensional obstacle problem for a uniformly elliptic, divergence form operator $L = div(A(x)\nabla)$ with Lipschitz continuous coefficients and discuss the optimal regularity of the solution. Our main result states that, similarly to what happens when $L = \Delta$, the variational solution has the optimal interior regularity. We achieve this by proving some new monotonicity formulas for an appropriate generalization of Almgren's frequency functional.
Bo Guan (Ohio State) on November 20, 2013
Title: The concavity and subsolution in estimates for fully nonlinear elliptic equations
Abstract: We report recent progresses in our effort of seeking methods to
derive a priori second order estimates for fully nonlinear elliptic equations
under general structure conditions. In this talk we shall focus on the Dirichlet
problem in Euclidean space. The main topic will be the role of concavity and
subsolutions.
Bo Guan (Ohio State) on Jan 22, 2014
Title: The Dirichlet Problem for Fully Nonlinear Elliptic Equations
Abstract: We report our recent work on fully nonlinear elliptic equations
on Riemannian manifolds. We introduce some new techniques in
deriving a priori estimates, which can be used to treat a variety of
types of fully nonlinear elliptic and parabolic equations on real or
complex manifolds. As a result we are able to solve the Dirichlet
problem in a domain with no geometric restrictions under optimal
structure conditions. The result is new and optimal in the Euclidean
case.
Feride Tiglay (Ohio State) on Feb 12, 2014
Title: Integrable Evolution Equations on Spaces of Tensor Densities
Abstract: We present how two nonlinear partial differential equations arise naturally as EulerArnold equations on spaces of tensor densities. These equations, like several other equations from mathematical physics that fit in the same framework, possess some hallmarks of integrability. We discuss the well posedness of the Cauchy problem and break down of solutions.
Avner Friedman (Ohio State) on Feb 19, 2014
Title: The Dynamics of Ant Trails
Abstract: Armies of ants are known to move in trails. These trails are formed by a chemotactic force induced by pheromone secreted by the ants. In this talk I shall introduce a mathematical model consisting of two partial differential equations, which explain when and how these trails are formed. The first equation, for the ants, includes the chemotaxis effect of pheromone and the dispersion caused by overcrowding. The second equation is a reactiondiffusion equation for the pheromone concentration. The strength of the chemotactic force, Ï‡, plays a critical role in the analysis. We prove that trails cannot be formed if Ï‡ is small, while many trails exist if Ï‡ is large.
Stephen Preston (Colorado) on Feb 26, 2014
Title: Higherdimensional HunterSaxton and CamassaHolm equations
Abstract: The HunterSaxton equation arises in a model of liquid crystals, while
the CamassaHolm equation arises in the study of water waves. Both
equations are onedimensional PDEs which are both completely
integrable and are geodesic equations on infinitedimensional manifolds.
I will discuss some of their basic properties along with some natural
generalizations of these equations to higher space dimensions.
Xiangwen Zhang (Columbia) on March 19, 2014
Title: A Proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R^3
Abstract: A classical uniqueness theorem of Alexandrov says that: if M and M' are two closed strictly convex C^2 surface in R^3 and satisfy f(k1,k2)=f(k1',k2'), at pots of M, M' with parallel normals, for some C^1 function f(y1,y2) with (D_{y1}f)(D_{y2}f)>0, then M is equal to M' up to a translation. We will talk about a new PDE proof for this theorem by using the maximal principle and weak unique continuation theorem of BersNirenberg. More generally, we prove a version of this theorem with certain weaker regularity assumption: the spherical hessians of the supporting functions for the corresponding convex bodies as Radon measures are nonsingular. This is a joint work with P. Guan and Z. Wang.
Yaping Wu (Capital Normal University, China) on March 20, 2014
Title: Steady States and Traveling Waves for SKT Competition Model with CrossDiffusion
Abstract: In this talk we shall be focused on a quasilinear reaction diffusion system with cross diffusion, which was first proposed by Shigesada, Kawasaki and Teramoto in 1979 for investigating the spatial segregation of two competing species under inter and intraspecies population pressures. I shall talk about some recent research progress on the existence and stability of nontrivial steady states and travelling waves for the SKT competition model with cross diffusion, which may correspond to some new pattern formation and wave phenomena induced by cross diffusion.
Marcus Khuri (SUNY Stony Brook) on April 2, 2014
Title: The massangular momentum inequality for asymptotically flat
and asymptotically hyperbolic initial data
Abstract: Consider axisymmetric initial data for the Einstein
equations, having two ends, one asymptotically flat or asymptotically
hyperbolic and the other either asymptotically flat or asymptotically
cylindrical.
Heuristic physical arguments lead to the following inequality
m\geq\sqrt{J} relating the total mass and angular momentum. Equality
should be achieved if and only if the data arise from the exrteme Kerr
spacetime. When the designated end is asymptotically flat, Dain
established this inequality (along with the corresponding rigidity
statement) when the data are maximal and vacuum, and subsequently
several authors have improved upon and extended these results. Here we
consider the general nonmaximal case in which the matter fields
satisfy the dominant energy condition, and introduce a natural
deformation back to the maximal case which preserves all the relevant
geometry. This procedure may then be used to establish the angular
momentummass inequality (and rigidity statement) in the general case,
assuming that a solution exists to a canonical system of two elliptic
equations. This is joint work with Ye Sle Cha.
When the designated end is asymptotically hyperbolic (modeling
asymptotically null slices in asymptotically Minkowski spacetimes),
similar results hold. This is joint work with Anna Sakovich.
Richard Wentworth (Maryland) on April 9, 2014
Title: The YangMills flow on Kaehler manifolds
Abstract: The fundamental work of Donaldson and UhlenbeckYau proves the the smooth convergence of the YangMills flow of stable integrable unitary connections on hermitian vector bundles over Kaehler manifolds. This was generalized by Bando and Siu to incorporate certain (singular) hermitian structures on reflexive sheaves. BandoSiu also conjectured what happens when the initial sheaf is unstable; namely, that the limiting behavior should be controlled by the HarderNarasimhan filtration of the sheaf. In this talk I will describe the solution to this question, which draws on the work of several authors.
Lu Xu (Chinese Academy of Sciences) on April 16, 2014
Title: BrunnMinkowski inequalities and related problems
Abstract: The classical BrunnMinkowski inequality is an inequality relating the volumes of convex bodies. In this talk, we will discuss this type of inequality with respect to eigenvalue functinal of Hessian operator. We also use this inequality to prove the symmetric result for overdetermined Hessian problem.
Yong Huang (Chinese Academy of Sciences) on April 17, 2014
Title: The Lp Minkowski problem
Abstract: In this talk, I will make a historical survey of the Minkowski problem and the Lp Minkowski problems. It is related with the existence, uniqueness of the MongeAmpere equations on the unit sphere. The continuity methods and Variational methods solving this problem is discussed.
Wei Zhou (Minnesota) on April 23, 2014
Title: On the regularity for the Dirichlet problem for degenerate Hessian equations
Abstract: We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the C^{1,1}boundary data, we establish the interior C^{1,1}regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach. If time permits, we shall also discuss the optimal interior C^{0,1}regularity of the viscosity solution to the Dirichlet problem for certain nonconvex degenerate Hessian equations.
This page is maintained by KingYeung lam.
Does the page seem familiar?