The Seventh Ohio River Analysis Meeting

Roza Aceska (Ball State): Scalability generated by iterative actions of operators.

We study systems of vectors of the form (1) {\(A^n\)g : g ∈ G, n = 0, 1, 2, . . . }, where A is a bounded operator on a Hilbert space H and G is a discrete set of vectors in H. Recent results have made it clear if (1) is a frame, complete Bessel system etc. This opens the question of frame scalability; some of the recent results and open questions are discussed. The problem originates from a sampling theory problem, where the unknown signal is recovered from its spatio-temporal samples

Ryan Berndt (Otterbein): The weighted Fourier inequality, polarity, and reverse Hölder inequality.

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex set and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Hölder class, is used to formulate the sufficient condition.

Tilak Bhattacharya (Western Kentucky): Existence of viscosity solutions of Trudinger's equation and long time asymptotics.

Bob Booth (North Carolina): Localized energy for wave equations with degenerate trapping, Part II.

When studying wave equations on differentiable manifolds, it is known that geodesic trapping necessitates a loss in standard local energy estimates. For non-degenerate hyperbolic trapping, the loss is logarithmic. For elliptic trapping, everything is lost except a logarithm. In our work, we consider a manifold with degenerate hyperbolic trapping, attaining an algebraic loss. This talk will utilize a first weaker estimate that is interesting in its own right. We will focus on the use of a WKB inspired analysis to attain the sharp result and prove sharpness via a quasimode construction. This is joint work with Hans Christianson, Jason Metcalfe, and Jacob Perry.

Josh Brummer (Kansas State): Bilinear operators with homogeneous symbols, smooth molecules, and Kato-Ponce inequalities.

We present a unifying approach to establish mapping properties for bilinear pseudodifferential operators with homogeneous symbols in the settings of function spaces that admit a discrete transform and molecular decompositions in the sense of Frazier and Jawerth. As an application, we obtain related Kato-Ponce inequalities. This is a joint collaboration with Virginia Naibo.

Alan Chang (Chicago): The Kakeya needle problem for rectifiable sets.

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is joint work with Marianna Csörnyei.

Vani Cheruvu (Toledo): Wavelet regularized solution of Laplace equation in an arbitrary shaped domain.

A numerical method for the solution of interior Dirichlet problem for the two-dimensional Laplace equation in an arbitrary shaped bounded domain is presented. This domain is embedded in a circular domain and exploiting the idea of analytic continuation, an inverse problem is formulated and solved for the boundary values of the unknown function on the circular domain. The ill-conditioning associated the inverse problems is handled by a wavelet regularization. Numerical results are presented.

Erin Compaan (Illinois): Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line.

In this talk we consider the regularity properties of the "good" Boussinesq equation on the half line. We discuss local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover, we show that the nonlinear part of the solution is up to smoother than the initial data. The method involves extending the problem to the real line and using a contraction argument there. A forcing term is used to enforce the boundary conditions. This approach allows us to apply powerful harmonic analysis methods, which a priori do not seem useful in the half-line setting, to the problem. The result improves that in [Himonas-Mantzavinos 2015], being the first result that constructs solutions below the L² space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line. This is joint work with Nikolaos Tzirakis.

Stephen Deterding (Kentucky): Bounded point derivations on R^p(X) and approximate derivatives.

Let X be a compact subset of the complex plane and let R₀(X) denote the set of all rational functions whose poles lie off X. The function space R(X) is the uniform closure of R₀(X). The space R^p(X) is an important function space that is closely related to R(X). For 1 < p < ∞, R^p(X) is the closure of R₀(X) in the L^p norm.

A bounded point derivation on R(X) at a point x₀ is a bounded linear functional D on R(X) such that D(fg) = D(f)g(x₀) + D(g)f(x₀) for all functions f, g belonging to R(X). It is known if there is a bounded point derivation on R(X) at x₀, then every function in R(X) has an approximate derivative at x₀. An approximate derivative is defined in the same way as the usual derivative except that instead of taking the limit in the difference quotient over all of X, the limit is taken over a subset of X with full area density at x₀. One can also define bounded point derivations for R^p(X), although a slightly different definition must be made.

In this talk we will introduce bounded point derivations on R^p(X) and discuss when the existence of a bounded point derivation on R^p(X) implies that every function in R^p(X) has an approximate derivative at x₀.

Dong Dong (Illinois): Hilbert transforms in a 3 by 3 matrix.

We provide background and main results about 8 variants of the classical Hilbert transform. Some key methods, connections to other fields, and recent development will also be reported.

Maxim Gilula (Michigan State): An analytic perspective on stability for oscillatory integrals in higher dimensions.

Given real analytic functions \(\phi, \psi:\mathbb{R}^d\to\mathbb{R}\), we consider scalar oscillatory integrals with phase \(\phi+s*\psi\) for small real s. It has been known since 1976 that the estimate of such an integral is not uppersemicontinuous in the real variable s in general (with respect to the oscillatory variable, due to Varchenko). However, the techniques used to develop such a conclusion were not robust enough to examine how exactly the oscillatory integral estimate changes as \(s\to 0\), but only determined that the estimates may be different asymptotically in the oscillatory variable.

Under the same nondegeneracy condition of the phase assumed by Varchenko when he showed stability does not hold in general, we examine some simple cases where we can predict exactly how the stability does not hold by considering any finite oscillatory variable instead of just considering the asymptotic behavior. We argue that in some sense stability *does* hold if we consider this perspective instead of the classical one. The Newton polyhedron plays an important role in determining the estimates of such integrals.

Fazel Hadaifard (Kansas): The global regularity of the 2D critical Boussinesq system with α > 2/3.

We examine the question for global regularity for the Boussinesq equation with critical fractional dissipation (α, β): α + β = 1. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying sata, as long as α > 2/3. This improves upon some recent works.

The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type change of variables which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by a new set of commutator estimates (in both negative and positive index Sobolev spaces).

Keaton Hamm (Vanderbilt): Regular families of kernels for nonlinear approximation.

Best N-term wavelet approximations for Triebel-Lizorkin smoothness spaces have been studied by DeVore, Jawerth, and Popov (Amer. J. Math. 1992) for many wavelet systems. Analogously, best N-term approximations from the space of Gaussian approximants given by \[ \mathcal{G}_N := \bigg\{ \sum_{j=1}^N a_j e^{-\frac{|x-x_j|^2}{\sigma_j^2}}: (a_j) \subset \mathbb{C}, (x_j) \subset \mathbb{R}^d, (\sigma_j) \subset (0,\infty) \bigg\} \] were considered by Hangelbroek and Ron (J. Funct. Anal. 2010). In each case, optimal rates of \(O(N^{-s/d})\) are obtained where s is the smoothness parameter of the Triebel-Lizorkin space and d is the dimension. We give sufficient conditions on a family of approximating kernels \((\phi_\alpha)_{\alpha \in A}\) such that the space \[ \Phi_N := \bigg\{ \sum_{j=1}^N a_j \phi_{\alpha_j}(x-x_j): (a_j) \subset \mathbb{C}, (x_j) \subset \mathbb{R}^d, (\sigma_j) \subset (0,\infty) \bigg\} \] yields the same approximation orders.

John Holmes (Ohio State): The compressible Euler equations in \(\mathbb R^2\).

We consider the Cauchy problem correspoding to the compressible Euler equations with data in the Sobolev space \(H^s(\mathbb R^2)\). Local in time well-posedness in the sense of Hadamard for the system is well known when s > 2. We improve upon the well-posedness results by showing that the continuity of the data-to-solution map is sharp. In particular, the data-to-solution map for this system is not uniformly continuous from any bounded subset of \(H^s\) to the solution space \(C([-T, T]; H^s)\).

Alex Iosevich (Rochester): On a bilinear analog of the spherical averaging operator.

The classical spherical averaging operator \(Af(x)=\int_{S^{d-1}} f(x-y) d\sigma(y)\), where σ is the surface measure on \(S^{d-1}\), maps \(L^p({\Bbb R}^d) \to L^q({\Bbb R}^d)\) if (1/p,1/q) is in the closed triangle with the endpoints (0,0), (1,1) and (d/d+1, 1/d+1). This is a classical result due to Littman and Strichartz. We are to consider a family of bilinear operators \[ B_{\theta}(f,g)(x)=\int f(x-y)g(x-\theta y) d\sigma(y),\] where \(\theta \in O_d({\Bbb R})\). We are going to determine a sharp range of exponents \(p,q,r \ge 1\) such that \(B_{\theta}: L^p({\Bbb R}^d) \times L^q({\Bbb R}^d) \to L^r({\Bbb R}^d)\). This operator is motivated by problems in Erdos combinatorics related to counting the number of times a given triangle congruence class may arise in a finite point set in \({\Bbb R}^d\). We shall also discuss applications to combinatorial geometry.

Ramesh Karki (Indiana - East): Optimal reconstruction of initial data in some evolutionary PDEs via finite discrete samplings.

Developing an approach introduced in a recent paper by DeVore and Zuazua [1], we show how to reconstruct optimally initial data, in a suitable Sobolev class, for some evolutionary PDEs, using only finite discrete measurements. In particular, we show that for our classes of PDEs the optimal sampling does not depend on the spectrum of the operators involved, but just on the order of the PDE, thus answering a question posed in [1]. We also tackle the same problem in the case in which the coefficients of the PDEs depend explicitly on time, thus generating a non-autonomous dynamical system. We finally comment on the possibility of using a variation of this approach to deal with some nonlinear integro-differential equations or non-linear PDEs that are C-integrable.

This is joint work with R. Aceska and A. Arsie.

Anton Lukyanenko (Michigan): Separated nets in nilpotent groups.

We generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We show that every such group G contains separated nets that are not biLipschitz equivalent. We define a class of separated nets in these groups arising from a generalization of the cut-and-project quasi-crystal construction and show that generically any such separated net is bounded displacement equivalent to a separated net of constant covolume. In addition, we use a generalization of the Laczkovich criterion to provide 'exotic' perturbations of such separated nets.

Lukáš Malý (Cincinnati): Neumann problem for p-Laplace equation in metric spaces.

I will discuss a variational approach to study the Neumann boundary value problem for the p-Laplacian on bounded smooth-enough domains in the metric setting, where Newtonian functions (defined via upper gradients) are to be utilized as a suitable counterpart of Sobolev functions. Fundamental questions of existence and uniqueness of solutions will be answered. Then, the De Giorgi inequality will be adapted to the Neumann problem in order to show boundedness and continuity (up to the boundary) of solutions for bounded/continuous Neumann boundary data.

Rajinder Mavi (Michigan State): Anderson localization with degenerate spectrum.

Random Schrodinger operators in strong disorder are known to obtain a phase, known as Anderson Localization, of pure point dense spectrum with exponentially localized eigenfunctions. An important characteristic of this phase is that in a properly renormalized setting the eigenvalue process approximates a Poisson process. We will investigate models with strong disorder where eigenvalues will approach a finite or infinite degeneracy. In such models we demonstrate a partial result in Anderson localization, and discuss the effect of tunneling in delocalizing the remaning portion of the wavepacket.

This talk is based on joint work with Jeffery Schenker.

Darío Mena Arias (Georgia Tech): Sparse operators and the sparse T1 theorem.

We impose standard T1-type assumptions on a Calderón-Zygmund operator T, and deduce that for bounded compactly supported functions f, g there is a sparse bilinear form \(\Lambda\) so that \[ \lvert \langle T f, g \rangle\rvert \lesssim \Lambda (f,g). \] The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) \(L ^{p}\) space.

Alex Mesiats (Courant): Convex duality in nonconvex variational problems.

We consider the minimization problem which models martensitic (diffusionless) phase transitions in rectangular domains. Physical experiments suggest that if opposite phases are present at opposite sides, the transition has a form of a zig-zag wall. My presentation addresses the mathematical model of this phenomenon, which involves the minimization of singularly perturbed 2D and 3D elastic energies with phase constraints. By means of sharp upper and lower bounds, we show that the experimentally observed zig-zag structure provides optimal energy scaling law. Despite the fact that the problem is highly nonconvex due to the presence of nonconvex phase constraints and the singular perturbation, in my talk, I will describe a relaxation method which allows to use the convex duality technique for the purpose of obtaining a sharp lower bound.

Sergii Myrochnychenko (Kent State): On the uniqueness of polytopes with congruent spherical projections.

We will discuss the problem of unique determination of convex bodies with congruent spherical projections. Consider two convex bodies K, L contained in the interior of a unit ball Bⁿ, n ≥ 3. Assume that for any point z on the unit sphere Sⁿ (a source of light), the spherical shadows K_z and L_z of the bodies are congruent (here K_z = {x ∈ S^n-1: [x,z] ∩ K ≠ ∅}). Does it follow that K=L? We give the affirmative answer to this question in the class of convex polytopes.

Andrea Nahmod (Massachusetts): Randomization and dynamics in nonlinear wave and dispersive equations.

In this talk we show how certain well-posedness results that are not available using only deterministic techniques (eg. Fourier and harmonic analysis) can be obtained when introducing randomization in the set of initial data and using powerful but still classical tools from probability as well. These ideas go back to seminal work by J. Bourgain on the invariance of Gibbs measures associated to dispersive PDE. We will first explain some of these ideas and review some recent probabilistic well-posedness results. We will then describe recent work of myself joint with Chanillo, Czubak, Mendelson and Staffilani in which we treat probabilistic well-posedness of a geometric wave equation with randomized supercritical data.

Bae Jun Park (Wisconsin): On the boundedness of pseudo-differential operators on Triebel-Lizorkin spaces.

I will talk about some endpoint boundedness properties of pseudo-differential operators in OP(\(\mathcal{S}_{\rho, \rho}^m\)), 0 < ρ < 1, on Triebel-Lizorkin spaces. It was proved that operators in OP(\(\mathcal{S}_{\rho, \rho}^m\)) are bounded in \(h^p\), 0 < p < ∞, and Sobolev spaces with some restriction on m. I extend these results to Triebel-Lizorkin spaces. The result is motivated by the boundedness of a radial multiplier m(D) = \(e^{-2\pi i|D|^{(1-p)}}(1+|D|^2)^{m/2}\), which is a typical example of operators in OP(\(\mathcal{S}_{\rho, \rho}^m\)).

Guanying Peng (Cincinnati): Regularity of the Eikonal equation with two vanishing entropies.

We study regularity of solutions to the Eikonal equation \(|\nabla u|=1\) a.e. in a bounded simply-connected two dimensional domain. With the help of two vanishing entropies, we prove that solutions of the Eikonal equation are locally Lipschitz continuous, except at a locally finite set of points in the domain. The motivation of our problem comes from the zero energy state of the Aviles-Giga functional in connection with the theory of smectic liquid crystals and thin film blisters. Our results for the first time use only two entropies to characterize regularity properties in this direction.

This is joint work with Andrew Lorent at the University of Cincinnati.

Jacob Perry (North Carolina): Localized energy for wave equations with degenerate trapping, Part I.

When studying wave equations on differentiable manifolds, it is known that geodesic trapping necessitates a loss in standard local energy estimates. For non-degenerate hyperbolic trapping the loss is logarithmic, while for elliptic trapping everything is lost except a logarithm. In our work, we consider an asymptotically flat manifold with degenerate hyperbolic trapping. This talk will develop a local energy estimate that holds everywhere except on the trapped set, via separate analysis of high and low frequency behavior. This estimate allows for local analysis near the trapped set, where algebraic loss can be attained. This talk is part of a joint work with Robert Booth, Hans Christianson, and Jason Metcalfe.

Dat Pham (Wayne State): Lipschizian and Hölderian full stability for parametric variational systems.

The paper studies the notions of Lipschitzian and Hölderian full stability of solutions to general parametric variational systems described via partial subdifferential and normal cone mappings acting in Hilbert spaces. We derive sufficient conditions of those full stability notions without assuming differentiability and computable formula for the modulus of prox-regularity of lower semicontinuous functions. The obtained results combining with related results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions.

This is joint work with Boris Mordukhovich (Wayne State University) and Nghia Tran (Oakland University).

Stefanie Petermichl (Toulouse): On the matrix A₂ conjecture.

The condition on the matrix weight W that is necessary and sufficient for the L²(W) boundedness of the Hilbert transform, the matrix A₂ condition, is known since 1997 (Treil-Volberg). Good or sharp quantitative norm control depending on the A₂ characteristic of the weight in the 'scalar' case dates to 2007 (P.). Despite notable improvement and many new techniques that apply in the scalar setting, the matrix A₂ question for the Hilbert transform remains unsolved. We present the best to date estimate (Nazarov-P.-Treil-Volberg), via the use of certain convex bodies. Certain maximal functions of Christ-Goldberg type already enjoy best estimates in the matrix setting. We present the first sharp estimate of a singular operator in this setting, the matrix weighted square function (Hytonen-P.-Volberg).

Stephen Quinn (Missouri): A sublinear Schur's test for elliptic PDEs.

The Schur's Test provides a condition for the boundedness of an operator based on existence of functions satisfying certain inequalities. We will use this concept to provide necessary and sufficient conditions for the existence of solutions to a class of Elliptic PDEs. The main restriction is that the Green's Kernel is quasi-symmetric and satisfies a Weak Maximum Principle. We connect our solutions to a weighted norm inequality. This is joint work with Igor Verbitsky.

Alexander Reznikov (Vanderbilt): Separation and covering properties of greedy energy points.

We discuss a method to generate a family (in fact, a sequence) of points on a unit sphere that are not asymptotically uniformly distributed; however, they have decent separation and covering properties. These points are generalization of Leja points that are used for numerical integration. We also discuss some open problems in this direction.

Diego Ricciotti (Pittsburgh): Regularity for p-harmonic functions in the Heisenberg group.

We provide a proof of the \(C^{1,\alpha}\) regularity of weak solutions to the p-Laplace equation in the Heisenberg group for p > 4.

Matthew Romney (Illinois): Geometry of Grushin space.

The Grushin plane is a standard example of a non-equiregular sub-Riemannian manifold. This talk will discuss quasiconformal and bi-Lipschitz mappings from the Grushin plane into Euclidean space, with emphasis on how these results relate to more classical aspects of quasiconformal mapping theory. We also discuss generalizations to a broader class of Grushin-type spaces. Portions of this talk are based on joint work with V. Vellis, C. Gartland, and D. Jung.

Chase Russell (Kentucky): Homogenization in perforated domains and interior Lipschitz estimates.

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale ε by establishing H¹-convergence rates for such solutions. The interior estimates are derived directly without the use of compactness. As a consequence, we derive a Liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains.

Luis San Martín Jiménez (UNAM): A connection between two initial Dirichlet-type problems for parabolic equations.

We study operators \(L=\text{div}(A(x,t)\nabla)-\partial_t\) where \(A(x,t)\) is a smooth and symmetric matrix satisfying the ellipticity condition \[\lambda|\xi|^2\leq\sum_{i,j=1}^{n}A_{ij}(x,t)\xi_i\xi_j \leq\lambda^{-1}|\xi|^2,\quad\quad\quad (x,t)\in\mathbb{R}^{n+1}, \quad\xi\in\mathbb{R}^n.\] We consider solutions to \(Lu=0\) in a Lipschitz cylinder \(\Omega_T\), and define a Regularity problem (R)_p in this setting. Finally, we prove that (R)_p implies (D)_p', where p and p' are conjugate exponents.

Morgan Schreffler (Kentucky): Approximation of solutions to the mixed problem on Lipschitz domains.

The mixed problem, or Zaremba's problem, has many physical applcations. In this talk we consider a method for approximating solutions to the mixed problem on a Lipschitz domain Ω. Further, we determine a desirable rate of convergence and attempt to formally differentiate with respect to the approximating parameter ε.

Mariana Smit Vega Garcia (Washington): The singular free boundary in the Signorini problem.

In this talk I will overview the Signorini problem for a divergence form elliptic operator with Lipschitz coefficients, and I will describe a few methods used to tackle two fundamental questions: what is the optimal regularity of the solution, and what can be said about the singular free boundary in the case of zero thin obstacle. The proofs are based on Weiss and Monneau type monotonicity formulas. This is joint work with Nicola Garofalo and Arshak Petrosyan.

Christopher Sogge (Johns Hopkins): On the concentration of eigenfunctions.

I shall present some results in global harmonic analysis that concern properties of eigenfunctions on compact Riemannian manifolds. Using local arguments we can show that L^p norms of eigenfunctions over the entire manifold are saturated if and only if there are small balls (if p is large) or small tubular neighborhoods of geodesics (if p is small) on which the eigenfunctions have very large L^p mass. Neither can occur on manifolds of nonpositive curvature, or, more generally, on manifolds without conjugate points.

Gareth Speight (Cincinnati): Porosity and differentiability in Euclidean spaces and Carnot groups.

Rademacher's theorem states that Lipschitz functions between Euclidean spaces are differentiable almost everywhere. Investigating validity of a converse to Rademacher's theorem leads to the construction of small universal differentiability sets, which contain points of differentiability for all Lipschitz functions. Porous sets are sets with relatively large holes on arbitrarily small scales, and have applications to the study of differentiability. For instance, a universal differentiability set cannot be a countable union of porous sets.

We discuss measure zero universal differentiability sets in the Heisenberg group and applications of porosity to differentiability in Carnot groups, which can be used to help prove Pansu's differentiability theorem for Lipschitz mappings.

Eric Stachura (Haverford): Existence of propagators for Coulomb-like potentials in density functional theory.

We prove existence of propagators for a time dependent Schrödinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential. Finally we show the new potentials are dilatation analytic, and so the spectrum of the corresponding Hamiltonian can be fully characterized.

Alex Stokolos (Georgia Southern): Harmonic analysis and cycles in non-linear dynamical systems.

I will address the problem of cycle detection and stabilization in non-linear discrete autonomous dynamical systems. This is a joint talk with D.Dmitrishyn, A.Khamitova, P.Hagelstein and M.Tohaneanu.

Dmitriy Stolyarov (Michigan State): Operator of integration on the space of bounded analytic functions.

The operator of integration on the space of bounded analytic functions on a simply connected domain \(\Omega\subset \mathbb{C}\) is bounded if and only if the interior diameter of \(\Omega\) is finite. I will prove this fact and show its relationship with the theory of generalized Volterra operators. This is joint work with Wayne Smith and Alexander Volberg.

Jeremy Tyson (Illinois): Quasiregular mappings of Euclidean space and the Heisenberg group.

I will discuss two lines of research related to quasiregular mappings. A continuous map f is quasiregular (QR) if it lies in a suitable (local) Sobolev space and satisfies a uniform relative analytic distortion condition. Roughly speaking, the singular values of the (a.e. defined) differential matrix Df should be comparable up to a fixed multiplicative constant. We allow the possibility that Df vanishes in a point. Quasiregular mappings are noninjective generalizations of quasiconformal mappings, and higher-dimensional analogs of holomorphic functions. Topologically, they are branched covers, in particular, they are local homeomorphisms off of a small branch set.

Higher-dimensional QR maps exhibit certain types of rigidity not shared by their planar counterparts. Specifically, QR maps whose distortion is sufficiently small, or which are sufficiently regular, are necessarily unbranched. I will survey the literature on such rigidity results in the Euclidean setting.

Second, I will discuss the QR ellipticity problem. A Riemannian n-manifold is quasiregularly elliptic if it receives a nonconstant QR map from \({\mathbb R}^n\). QR ellipticity will be discussed both for mappings \(f:{\mathbb R}^n \to N\) into a Riemannian n-manifold, and mappings \(f:{\mathbb H}^n \to N\) into a contact sub-Riemannian manifold. Here \({\mathbb H}^n\) denotes the Heisenberg group, which we equip with its standard sub-Riemannian metric structure. The theory of QR mappings between sub-Riemannian manifolds is in its infancy, and the Heisenberg QR ellipticity problem is a good testbed to try out various definitions of sub-Riemannian quasiregularity.

Vladimir Vinogradov (Ohio University): On applications of special functions in probability theory.

We employ numerous analytical methods and special functions for solving problems of probability theory. Some of our results illustrate the widely recognized fact that probability theory is an excellent source of new problems for analysis. We demonstrate how such interplay between these two disciplines is effective in providing a simple derivation of new assertions of probability theory, as well as in finding a meaningful interpretation of particular analytical results, or even in deriving previously unknown representations for some special functions.

This is joint work in collaboration with Richard Paris (Abertay University).

Dekai Zhang (Minnesota): The Neumann problem of Hessian quotient equations.

We obtain global \(C^2\) apriori estimates and thus prove the existence theorem of the Neumann problem of Hessian quotient equations. This is a recent joint work with Chuanqiang Chen.

Jinping Zhuge (Kentucky): Periodic homogenization of elliptic equations with oscillating boundary data.

In this talk, we will consider the homogenization of elliptic system in divergence form with both periodic oscillating coefficients and boundary data (joint work with Zhongwei Shen). We obtained the nearly sharp rate of convergence O(ε^1/2-) (improving to O(ε^1/4-) if d = 2) in strictly convex domains for both Dirichlet and Neumann problems. An algebraic rate of convergence was also obtained in the domains of finite type.

Scott Zimmerman (Pittsburgh): Sobolev extensions of Lipschitz mappings into metric spaces.

Wenger and Young proved that the pair \((\mathbb{R}^m,\mathbb{H}^n)\) has the Lipschitz extension property for m ≤ n where \(\mathbb{H}^n\) is the sub-Riemannian Heisenberg group. That is, for some C > 0, any L-Lipschitz map from a subset of \(\mathbb{R}^m\) into \(\mathbb{H}^n\)can be extended to a CL-Lipschitz mapping on \(\mathbb{R}^m\).

In this talk, I construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension m. I will show that any Lipschitz mapping from a compact subset of \(\mathbb{R}^m\) into \(\mathbb{H}^n\) may be extended to a Sobolev mapping on any bounded domain containing the set. More generally, I will explain this result in the case of mappings into any Lipschitz (n-1)-connected metric space.