The Third Ohio River Analysis Meeting

University of Cincinnati | March 9-10, 2013 | Cincinnati, Ohio

Talk Abstracts: Invited Addresses are indicated in green.

Luis Acuña (Purdue): Trace asymptotics for fractional Schrödinger operators.

An analogue of a result on the asymptotic expansion for the trace of Schrödinger operators on R^d is proved when the Laplacian Δ, which is the generator of the Brownian motion, is replaced by the non-local integral operator $Δ^{(α / 2)}$ , 0 < α < 2, which is the generator of the symmetric stable process of order α. Some extensions to Schrödinger operators arising from relativistic stable and mixed stable processes are obtained.

Murat Akman (Kentucky): On the dimension of a certain measure in the plane.

We will discuss the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. This is a joint work with John L. Lewis.

Rodrigo Bañuelos (Purdue): Fourier multipliers arising from Lévy processes.

We will discuss recent applications of martingale inequalities to Fourier multipliers that arise from transformations of Lévy processes via the Lévy–Khintchine formula. While these results are of interest on their own right, they are motivated by applications to some classical multipliers such as Riesz transforms, the Beurling-Ahlfors operator, Marcinkiewicz multipliers, and connections to rank-one convexity and quasiconvexity.

Some background material can be found in the speaker's overview article "The foundational inequalities of D. L. Burkholder and some of their ramifications," Illinois J. Math. Volume 54, Number 3 (2010), 789-868.

Marius Beceanu (IAS): Dispersive estimates for equations with time-dependent potentials.

We prove dispersive estimates for the Schrödinger and wave equations with a wide class of time-dependent potentials.

Thomas Bieske (South Florida): Generalizations of a Laplace-type equation in the Heisenberg group and class of Grushin-type spaces.

Beals, Gaveau and Greiner (1996) found the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces (Bieske-Gong, 2006) and the Heisenberg group (Capogna, Danielli, Garofalo, 1997). We extend the 2-Laplace-type equation to a p-Laplace-type equation. We show that the obvious generalization does not have desired properties, but rather, our generalization does preserve some natural properties.

Dat Cao (Missouri): Existence and pointwise estimates of solutions to subcritical quasilinear elliptic equations.

In this talk, we study the following problem:

-Δ_pu = σ u^q in Rⁿ, $inf_{x \in R^{n}}$ u(x) = s, s ≥ 0,
where σ is a locally finite nonnegative measure, Δ_p is the p-Laplacian, 1 < p < n and 0 < q < p-1. We will present the global pointwise estimates for the solutions of the above equation in terms of Wolff's potential. This is joint work with Igor E. Verbitsky.

Tanya Christiansen (Missouri): Counting resonances for Schrödinger operators.

Physically, a resonance may correspond to a decaying wave. This is in contrast with an eigenvalue of a selfadjoint operator, which in many models corresponds to a periodic wave. Mathematically, resonances are analogs of discrete spectral data on certain noncompact domains or manifolds. Their behavior is less well understood than that of eigenvalues of selfadjoint operators, leaving many questions unanswered.

We give an introduction to resonances and the problem of counting them, concentrating on the case of Schrödinger operators on R^d. A subtext of the talk is that Schrödinger operators with complex potentials both provide intriguing examples and, in some settings, can facilitate proofs about Schrödinger operators with real-valued potentials.

David Freeman (UC Blue Ash): Bi-Lipschitz and quasihomogeneous parameterizations.

In this talk we survey a few recent results providing necessary and sufficient conditions for a metric space to possess a bi-Lipschitz parameterization. We focus on certain metric spaces that admit a transitive uniformly bi-Lipschitz group action and demonstrate how tools from the theory of quasiconformal groups and strong A-infinity weights can be used to prove the existence of a bi-Lipschitz parameterization. We also explore the equivalence between bi-Lipschitz and quasihomogeneous parameterizations in the setting of Ahlfors regular metric spaces.

William Green (Rose-Hulman): Exponential decay for dispersion managed solitons for general dispersion profiles.

Consider the one-dimensional non-linear Schrödinger equation with periodically varying dispersion coefficient d(t),

i u_t + d(t) u_xx + |u|²u = 0
This equation can be used as a model for the amplitude of a signal transmissitted along a carrier wave in fiber optic cables. We show that stationary solutions of an averaged version and their Fourier transforms of this equation decay exponentially. This is joint work with Dirk Hundertmark.
Robert Hardt (Rice): Some homology and cohomology theories for a metric space.

Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space, completed to get flat chains, and defined flat cochains as the dual space. Federer and Fleming also considered these in the sixties and seventies, for homology and cohomology of Euclidean Lipschitz neighborhood retracts. These include smooth manifolds and polyhedra, but not algebraic varieties or subspaces of some Banach spaces. In works with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate topologies for flat chains and cochains in general metric spaces. With these, we homologically characterize Lipschitz path connectedness and obtain several facts about spaces that satisfy local linear isoperimetric inequalities.
Jarod Hart (Kansas): A multilinear local T(b) theorem for square functions.

This joint work in harmonic analysis with L. Oliveira and A. Grau de la Herrán addresses the study of oscillatory behavior of functions in the context of multilinear operators. In particular, we introduce a local testing condition, in place of typical global testing conditions, that is sufficient for bounds of some Littlewood-Paley type square functions. Given the local conditions on the square function operator, Carleson measure techniques are used to obtain L² estimates. These square function estimates are applied to give a new local testing condition sufficient for multilinear Calderón-Zygmund operators as well.
Jay Hineman (Kentucky): Well-posedness of nematic liquid crystal flow in $L_{uloc}^{3}$ (R³).

We discuss the local well-posedness of the Cauchy problem for a simplified version of hydrodynamic flow of nematic liquid crystals in R³ for any initial data (u₀, d₀) having small $L_{uloc}^{3} (R^{3})$ -norm of (u₀, ∇d₀). Here $L_{uloc}^{3} (R^{3})$ is the space of uniformly locally L³-integrable functions.
Steve Hofmann (Missouri): Recent progress on the theory of boundary value problems for divergence form elliptic equations.

Benjamin Jaye (Kent State): The fractional Riesz transform and an exponential potential.

We shall describe some joint work with Fedor Nazarov and Alexander Volberg, concerning the geometric properties of general measures for which an associated singular integral operator (of non-integer dimension) is bounded.
Nguyen Lam (Wayne State): Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications.

Sookkyung Lim (Cincinnati): Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method.

Flagellar bundling is an important aspect of locomotion in bacteria such as Escherichia coli. To study the hydrodynamic behavior of helical flagella, we present a computational model that is based on the geometry of the bacterial flagellar filament at the micrometer scale. We consider two model flagella, each of which has a rotary motor at its base with the rotation rate of the motor set at 100 Hz. Bundling occurs when both flagella are left-handed helices turning counterclockwise (when viewed from the nonmotor end of the flagellum looking back toward the motor) or when both flagella are right-handed helices turning clockwise. Helical flagella of the other combinations of handedness and rotation direction do not bundle.
Qing Liu (Pittsburgh): Horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group.

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. We show a comparison principle in the axisymmetric case. Our existence theorem is based on a game-theoretic approach. An explicit solution is also given for the motion starting from a subelliptic sphere. In addition, we present several properties of the mean curvature flow in the Heisenberg group. This is joint work with Fausto Ferrari and Juan Manfredi.
Galyna Livshyts (Kent State): Maximal surface area of a convex set in Rⁿ with respect to exponential rotation invariant measures.

Let p be a positive number. We consider the probability measure γ_p with density $φ_{p} (y) = c_{n, p} e^{({- | y |}^{p} / p)}$ . We show the optimal asymptotic bound for the maximal surface area of a convex body in Rⁿ with respect to γ_p, which is a generalization of results due to Ball and Nazarov in the case of the standard Gaussian measure γ₂.
Michael Music (Kentucky): Exceptional circles of radial potentials.

We study singularities in a nonlinear scattering transform for the two-dimensional Schrodinger equation at zero energy. We look at radial perturbations of conductivity type potentials to determine when singularities arise in this scattering transform. For small perturbations, an explicit description of the set of singularities will be given. This is joint work with Peter Perry and Samuli Siltanen.
Nathan Pennington (Kansas State): Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces.

Through the use of a non-standard Leibniz rule estimate, we prove the existence of unique local and global solutions to the incompressible isotropic Lagrangian Averaged Navier-Stokes equation with initial data in various categories of Besov spaces. Specifically, for p > n, we get local existence with initial data u₀ ∈ $B_{p, q}^{r}$ (Rⁿ) for r > 0. For p = 2, we get local existence with initial data u₀ ∈ $B_{2, q}^{n / 2 - 1}$ (Rⁿ) and the local solution can be extended to a global solution for n = 3, 4.
Alexei Poltoratski (Texas A&M): Gelfand-Levitan theory for the Dirac equation.

The classical Gelfand-Levitan theory translates spectral problems for differential operators into the language of complex and harmonic analysis. In my talk I will discuss the G-L theory in relation with the Krein - de Branges theory of Hilbert spaces of entire functions, along with some new proofs for the classical theorems on the spectral analysis of the Dirac equation.
Surya Prasath (Missouri): Global dissipative solutions for generalized forward-backward diffusion equations.

We consider a generalized nonlinear parabolic diffusion equation arising in image processing problems. Based on the well-known Perona-Malik equation, a coupled PDE is derived by introducing a constraint on the original diffusion coefficient. Unlike other classical forward-backward diffusion problems which are known to be ill-posed, this Dirichlet initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions). We provide several properties of these solutions as well. This is a joint work with Dmitry Vorotnikov (Univ. Coimbra, Portugal) and Jose A. I. Martinez (Technion, Israel).
Alexander Reznikov (Michigan State): One sided bump conditions and weak and strong two weight boundedness of Calderón-Zygmund operators.

During last one and a half years the "Bump conjecture" was proved with two different methods, and the "T1 theorem" for Hilbert transform was verified. One left open question is the following: is the one sided bump (sufficient for boundedness of Maximal function) sufficient for weak L²-boundedness of the Hilbert transform?

We discuss the case when the answer is positive.
Rishika Rupam (Texas A&M): Boundary behavior of meromorphic inner functions on the upper half plane.

We will introduce and use tools of Clark measures to characterize solutions of a long standing problem first studied by Louis de Branges, namely: for what sequences {a_i} on R do there exist meromorphic inner functions on the upper half plane, whose spectrum is exactly the set {a_i} and whose derivative is uniformly bounded on R?
Aaron Saxton (Kentucky): Exponential decay of the resolvent of a Schrödinger operator in the trace norm.

In 1973, Combes and Thomas discovered a general technique for showing exponential decay of eigenfunctions. The technique involved proving the exponential decay of the resolvent of the Schrödinger operator localized between two distant regions. Since then the technique has been been applied to several types of Schrödinger operators culminating in a paper by Germinet and Klein presenting optimal estimates in the operator norm. In my talk, I will review the basic techniques and extend them to trace class estimates on the localized resolvent of a Schrödinger operator.
Armin Schikorra (Planck Institute): Knot-energies and fractional harmonic maps.

We will present a proof that curves (knots) which are stationary for the Moebius energy are smooth in the critical dimension. This energy was introduced by O'Hara (Topology, 30(2):241–247, 1991), and it was shown by Freedman, He and Wang (Ann. of Math. (2), 139(1):1–50, 1994), and He (Comm. Pure Appl. Math., 53(4):399–431, 2000) that minimizers of this energy are smooth, using a geometric argument employing crucially the Moebius invariance of this energy.

Our approach, however, which shows regularity even for critical points, does not rely on this geometric invariance, but rather uses analytic methods from Harmonic Analysis and Potential Theory inspired by the regularity arguments of fractional harmonic maps by Da Lio-Riviere (APDE, 4(1):149–190, 2011; Advances in Mathematics, 227:1300–1348, 2011) and the speaker (J. Differential Equations, 252:1862–1911, 2012; Preprint, arXiv:1103.5203, 2011).

This is joint work with S. Blatt and P. Reiter.
Alex Stokolos (Georgia Southern): On the rate of a.e. convergence of certain classic integral means.

We are going to present several nearly optimal results concerning the rate of almost everywhere convergence of the Fourier integral means T_t f. A typical result for these means is the following:

If the function f belongs to the Besov space $B_{p, p}^{s}$ in the range 1 < p < ∞, 0 < s < 1, then T_t f(x) − f(x) = o_x(t^s) a.e. as t→0⁺.

The above statement is valid for the Abel-Poisson, Gauss-Weierstrass, Bochner-Riesz means in particular. This is joint work with Walter Trebels.
Bing-Yu Zhang (Cincinnati): Nonhomogeneous boundary value problems of the nonlinear Schrödinger equation.

It is well known that the Cauchy problem of the nonlinear Schrödinger equation

i u_t + u_xx + λ|u|²u = 0 u(x,0) = φ(x)
posed either on the real line R or the torus T is well-posed in the classical Sobolev space H^s for any s≥0. In this talk we will consider the equation posed either on the half line R⁺ = (0,∞) with nonhomogeneous boundary condition u(0,t) = h(t) or on a finite interval (0,L) with the nonhomogeneous boundary conditions u(0,t) = h₁(t), u(L,t) - h₂(t).

We will demonstrate that the initial-boundary-value problem (IBVP) of the nonlinear Schrödinger equation, when posed on the half line R⁺, is well-posed in the space H^s(R⁺) for any s≥0 with φ ∈ H^s(R⁺), h ∈ $H_{loc}^{\frac{2 s + 1}{4}}$ (R⁺).

The IBVP of the equation when posed on the finite interval (0,L) is well-posed in the space H^s(0,L) for any s≥0 with φ ∈ H^s(R⁺), h₁, h₂ ∈ $H_{loc}^{\frac{s + 1}{2}}$ (R⁺).

The results reported in this talk are joint work with Jerry L. Bona of University of Illinois at Chicago and Shuming Sun of Virginia Tech.

The Ohio River Analysis Meeting is a joint project of the Universityof Cincinnati Department of Mathematical Sciences and the University of Kentucky Department of Mathematics.

It is made possible by additional generous support from the University of Cincinnati, the Charles Phelps Taft Research Center, and the National Science Foundation.