Third Prairie Analysis
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First Prairie Analysis Seminar

Second Prairie Analysis Seminar

Tadeusz Iwaniec, Syracuse University.

ABSTRACT: The central theme running through the 2003 Prairie Lectures is the class of nonlinear PDEs whose prototype is the $p$-\textit{harmonic equation}

$$ \textrm{div}|\nabla u|^{p-2}\nabla u= \textrm{div}f,     1 < p < \infty $$

A wider and more unifying framework will be established by means of differential forms and elliptic complexes, within which an associated $q$-harmonic \textit{dual system} will exist, $p+q =pq$. The selection of topics reflects years of our interaction with scholars and scientists in several disciplines of mathematics: geometric function theory, calculus of variations, nonlinear elasticity, composites with micro-structures, and so forth. Thus, $p$-\textit{harmonic analysis} will tell us something about the existence of elastic deformations, their regularity and topological behavior. While the Sobolev space $ W^{1,p}(\Omega)$ is considered the natural domain of the $p$-harmonic operator, we shall depart from this space and move into the realm of so-called \textit{very weak solutions} where important new applications lie. Viscosity solutions of the $\infty$-Laplacian will emerge in the lecture of Juan Manfredi. Participants may wish to consult with John Lewis about the parabolic case as well. Our study of the $p$-harmonic equations beyond their natural domain of definition brings us closer to new methods of PDEs such as \textit{nonlinear commutators} of singular integrals, \textit{Hodge decomposition} and much more. One might possibly say that the $p$-harmonic analysis is an excellent source of ideas upon which we build a viable theory of "nonlinear singular integrals", like Riesz transforms for the Laplacian. We shall face the truly fascinating task of capturing the fundamental role of the $p$-harmonic operators in nonlinear problems, a virtue Riesz transforms do not possess. However, we shall not leave behind the linear theory of singular integrals. Every effort will be made to reduce to a minimum the technical aspects in the interest of mathematical insights.

John Lewis, University of Kentucky.

ABSTRACT: In this talk we discuss very weak overdetermined boundary conditions for positive solutions to some elliptic partial differential equations of $ p$ Laplacian type in a bounded domain $ D. $ After a brief survey of related problems we outline recent work which shows these conditions imply uniform rectifiability of $ \partial D $ and also that they yield the solution to certain symmetry problems.

Juan Manfredi, University of Pittsburgh.

ABSTRACT: We will discuss several notions of weak solutions of the $p$-Laplace equation, including the case $p=\infty$, in $\mathbb{R}^n$ and in the Heisenberg group $\mathcal{H}$. Bieske's extension of two uniqueness theorems of R. Jensen to the Heisenberg group will be presented. Finally, we will address the regularity of $p$-harmonic functions in the Heisenberg group, including Cordes estimates recently developed in collaboration with Andr\'as Domokos.

1. Artem Zvavitch, University of Missouri-Columbia.

ABSTRACT: In this talk we will present a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian Measure of its sections. Using this formula we will solve the Busemann-Petty problem for Gaussian Measures, asking whether symmetric convex sets with smaller Gaussian Measures of hyperplane sections necessarily have smaller Gaussian Measure.

2. Roger W. Barnard, Texas Tech University.

ABSTRACT: For any two points a and b in the open disk U on the complex sphere S, let K be a curve separating a from b on S which splits S into two complimentary regions A containing a and B containing b, and let k be the part of this curve lying in the closure of U. In this talk we discuss our solution to the problem of how small can the average harmonic measure [w(a,k,A)+w(b,k,B)]/2 be by applying reduced modules and a numerical technique suggested by an idea of R.Kuhnau's. This problem can be inerpreted as a way to determine the minimal average temperture at two points of a long cyclinder composed of two media separated by a membrane each of which contains a reference point.

3. Brock Williams, Texas Tech University.

ABSTRACT: Circle packing has become an important tool for approximating conformal maps between planar regions. The fundamental idea is that a map between two circle packings with the same underlying triangulation must be approximately conformal. Now suppose we are given a triangulated physical surface. We can easily create a circle packing with the same triangulation as the physical surface and produce a map from the surface to the packing. This process has been used successfully by Bowers, Hurdal, Stephenson, et al, to create maps of the human brain. However, such a map will not be conformal since it was created using a packing and physical data instead of two circle packings. To create a conformal map, we must manipulate the packing to produce angles which match the physical data. We will describe this entire construction, including how to match angles using a discrete version of conformal welding.

4. Sergei Merenkov, University of Michigan.

ABSTRACT: Given two complex manifolds with algebraically isomorphic semigroups of holomorphic endomorphisms, when are the manifolds biholomorphically equivalent? The answer to this question is far from complete. I will present partial results when both manifolds are bounded domains in $C^n$, or when one of the manifolds is $C^n$. There are pairs of non-homeomorphic domains with isomorphic semigroups.

5. Peter Hasto, University of Michigan

ABSTRACT: In recent years there has been a surge of interest in Lebesgue and Sobolev spaces where the exponent is allowed to vary. This spaces are useful for modeling certain phenomena in non-homogeneous materials. In this expository talk I will discuss some basic properties of these spaces as well as recent advances. Several open questions will also be presented.

6. Luigi D'Onofrio, University of Napoli and Syracuse University

ABSTRACT: To every vector field $f\in \textbf{L}^q(\Omega,\mathbb R^n),\; \Omega\subset \mathbb R^n$, there corresponds a unique solution $\;u\in \textbf{W}^{1,p}_0(\Omega)\;$ of the $p$-harmonic equation $$ \text{div}\,|\nabla u|^{p-2}\,\nabla u \;=\; \text{div} f\;, $$ where the exponents satisfy H\"{o}lder's relation: $\,p\,,q > 1\,$ and $\,p+q=p\cdot q\,.$ The $\,p$-harmonic transform assigns to $\,f\,$ the gradient of the solution. $$ \mathcal H_p : \textbf{L}^q(\Omega,\mathbb R^n)\longrightarrow \textbf{L}^p (\Omega,\mathbb R^n) \;,\;\;\;\;\;\;\;\; \mathcal H_p\,(f) \;=\;\nabla u $$\\ More general PDEs and the corresponding nonlinear transforms are also considered.\\ We are concerned with the continuous extension of the $\,p$-harmonic transform beyond this so-called natural domain of definition. Namely, $$ \mathcal H_p : \textbf{L}^{\lambda q}(\Omega,\mathbb R^n)\longrightarrow \textbf{L}^{\lambda p}(\Omega,\mathbb R^n)\;,\;\;\;\;\text{for some parameters}\;\;\lambda \geq\; \text{max}\text{\tiny{$\left\{\frac{1}{p}\,,\,\frac{1}{q}\right\}$}} $$\\ First, we establish an \textit{Interpolation Theorem}. Because of nonlinearity, this result requires substantial innovations of the familiar Marcinkiewicz ideas from the linear theory. Then we explicitly identify the so-called \textit{critical parameter} $\,\lambda\,$ for which the existence, uniqueness and continuity of $\,\mathcal H_p\,\,$ take place. Surprisingly, the uniqueness property in unbounded domains is lost when $\,\lambda\,$ exceeds the critical parameter. It is a little more surprising that the $\,n$-harmonic transform in unbounded domains, say $\,\Omega =\mathbb R^n,\,$ cannot be extended to any Lebesgue space $\,\textbf{L}^{s}(\mathbb R^n,\mathbb R^n),\;$ with $\,s > n.\,$ In other words, the critical parameter is equal to 1 in this case.

7. James Peirce, UC Davis.

ABSTRACT: The Lagrangian averaged Navier-Stokes Equations are a coupled system of PDEs for the mean velocity field and Lagrangian covariance tensor designed to capture the dynamics of the Navier-Stokes equations at length scales larger than a parameter \alpha, while averaging the motion at scales smaller than \alpha. I will provide a short introduction to the anisotropic Lagrangian averaged Navier-Stokes equation, briefly review previous analytical results, and outline the proof of local-in-time well-posedness of solutions to these equations when the viscosity term is of a certain form. In addition, with time remaining, I will present numerical solutions to the equations assuming steady channel flow in the two cases of no-slip and inhomogeneous boundary conditions. These solutions yield an interesting result. Although the velocity is steady, the covariance tensor is not steady and in fact blows up near the boundary.

8. Caroline Sweezy, New Mexico State University.

ABSTRACT: There are subspaces of $BMO(\mathbb{R}^{n})$, $BMO(r)$, $1\leq r<\infty $, introduced in [S] and defined by the growth condition, \begin{equation*} \sup_{1\leq p < \infty }\left\{\frac{1}{p^{1/r}}\left( \sup_{Q}\left( \frac{1% }{\left\vert Q\right\vert }\int_{Q}\left\vert f(x)-f_{Q}\right\vert ^{p}dx\right)^{1/p}\right) \right\} \leq C_{0} < \infty \end{equation*} These spaces are shown to have rearrangement invariant hulls that are similar to the space weak-$L^{\infty }$, which was defined by Bennett, DeVore and Sharpley. It is proved that the Hardy-Littlewood maximal operator, if it is finite a.e., takes BMO(r) into itself, with norm boundedness.

9. Mikil Foss, Kansas State University.

ABSTRACT: Given a body of elastic material, a basic problem in elastostatics is to find a deformation of this body that displaces the body's surface in some prescribed manner and minimizes a given energy functional. To preclude, as possible minimizers, those deformations that reverse the orientation of the material in a part of the body or compress a part to a region with zero volume, it is physically reasonable that the energy functional be equal to infinity for these types of deformations. In this talk, I will present a condition on a minimizer of a physically reasonable energy functional that is sufficient for ensuring the minimizer is partially regular.

10. Leonid Kovalev, Washington University in St. Louis

ABSTRACT:The talk concerns the behavior of solutions of the stationary one-dimensional Schrödinger equation under certain rearrangements of its potential. First results of this kind were obtained in the 1970-80's by Matts Essén in connection with estimates of harmonic measure and the growth of subharmonic functions. I am going to describe a unified approach to this problem, which is based on the analysis of permutations of state transition matrices.

11. Virginia Naibo, University of Kansas.

ABSTRACT: We prove pointwise inequalities for the maximal operator over all the directions in $\mathbf{R}^n$ when acting on $l^q$-radial functions and on product functions. From these inequalities we deduce boundedness results on $L^p$ for $p > n$; these can be applied to other operators, in particular to the Kakeya maximal operator. This is joint work with Javier Duoandikoetxea.

12. Genevra Neumann, Kansas State University.

ABSTRACT: The behavior of a complex-valued planar harmonic function can be vastly different from that of an entire function. For example, an analytic polynomial takes every value a finite number of times. Harmonic polynomials can take a given value an infinite number of times and can exclude regions of the complex plane (even when no values are taken an infinite number of times.) Open questions concerning the number of distinct zeros of harmonic polynomials lead to questions concerning the valence of harmonic functions. In this talk, some of the previous work on valence of harmonic functions will be described and some recent results will be presented.

13. Chiara Frosini, Universita' di Firenze, Italy.

ABSTRACT: Let $\Delta^{n}$ be a unit polydisc in $\C^{n}$ and let $f$ be a holomorphic self map of $\Delta^{n}$. In case $n=1,$ it is well known, by Schwarz's lemma, that $f$ (not the identity) has at most one fixed point in the unit disc, $\Delta$. Moreover, if $f$ has one fixed point in $\Delta,$ then any (hyperbolic) disc centered at such a point is mapped into itself by $f.$ If $f$ has no fixed points in $Delta,$ there exists a unique boundary point, say $x\in\partial\Delta$, such that every horocycle $E(x,R)$ of center $x$ and radius $R>0$ (the boundary relative of hyperbolic disc) is sent into itself by $f$. Such a point is called the \emph{Wolff point of $f.$} In this talk we propose a definition of Wolff points for holomorphic maps defined on a (not necessarily strongly convex) bounded domain of $\C^{n}$.In particular we characterize the set of Wolff points, $W(f),$ of a holomorphic self-map $f$ of the polydisc in terms of the properties of the components of the map $f$ itself. Moreover we study the relationship between $W(f)$ and the target set of $f,\;\Gamma(f),$ namely the set of the limit points of the sequence of the iterates of $f$.

14. Gerard Ornas, McNeese State University.

ABSTRACT:In this talk, we use a method of Barnard, based on Julia Variations to solve an extremal problem for hyperbolically convex functions. A hyperbolically convex function can be viewed as a univalent analytic function sending the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ in to a hyperbolically convex subset of $\mathbb{D}$. A subset of $\mathbb{D}$, call it $B$, is hyperbolically convex provided the hyperbolic geodesic connecting any two distinct points in $B$ lies entirely in $B$. We denote by $H$ the class of hyperbolically convex functions normalized by fixing the origin. The extremal problem we solve is maximizing $Lf$ over the class $H$. $L$ is a functional of the form $\Re \left( \Phi \left( \log \frac{f'(z)}{f'(0)} \right) \right)$, with reasonable conditions on $\Phi$. This class includes, by choosing $\Phi = \exp (z)$, the functional $|f'(z)|\|f'(0)|$. As a result, we achieve a broad generalization of a recent result of Pommerenke, Mej{\'{i}}a, and Vasilieyev, using an entirely different technique. This technique also shows promise in addressing many problems, including a related open conjecture by Pommerenke {\emph{et al.}}

15. Robert Smits, New Mexico State University.

ABSTRACT: In this talk we will discuss old and new results for the heat kernel in some unbounded domains. I will talk about various relationships between the geometry of the set and the decay rate for the exit time of Brownian Motion. In specific cases exact results can be given and I will discuss these as well as point to connections between the decay rate and Martin Kernel.

16. Christian Wolf, Wichita State University.

ABSTRACT: We discuss the existence of ergodic measures of maximal Hausdorff dimension for hyperbolic sets of surface diffeomorphisms. This is a dimension-theoretical version of the existence of ergodic measures of maximal entropy. The crucial difference is that while the entropy map is upper-semicontinuous, the map $\nu\mapsto\dim_H\nu$ is neither upper-semicontinuous nor lower-semi\-continuous. This forces us to develop a new approach, which is based on the thermodynamic formalism. Remarkably, for a generic diffeomorphism with a hyperbolic set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures. This is a joint work with Luis Barreira.

17. Ivan Blank, University of Louisville.

ABSTRACT: We show a method to eliminate a type of mixed asymptotics in certain free boundary problems and give two examples of its application. It appears that these problems cannot be handled by the monotonicity formula of Alt, Caffarelli, and Friedman (1984), or by the more recent monotonicity formula of Caffarelli, Jerison, and Kenig (2002).

18. Diego Maldonado, University of Kansas.

ABSTRACT: In the 90's, L. Caffarelli introduced a geometric approach in the study of convex solutions to the equation $\det D^2 \varphi = \mu$, when the measure $\mu$ satisfies a doubling condition. In this talk we will mention the main aspects of this approach and some of our contributions concerning regularity, growth, and real analysis related to the solutions. This is joint work with Liliana Forzani.

19. Byung-Geun Oh, Purdue University.

ABSTRACT: For a compact set $E \subset \mathbb{C}$ containing more than two points, we study asymptotic behavior of normalized zero counting measures $\{ \mu_k \}_0^\infty$ for the derivatives of Faber polynomials associated with $E$. For example if $E$ has empty interior, we prove that $\{ \mu_k \}_0^\infty$ converges in the weak-star topology to a measure whose support is the boundary of $g(D)$, where $g : \{ |z| > 1 \}\cup \{\infty\} \to \overline{\mathbb{C}} \backslash E$ is a universal covering map such that $g(\infty) = \infty$ and $D$ is the Dirichlet domain (associated with $g$) with center at $\infty$. Our results are counterparts of those of Kuijlaars and Saff (1995) on zeros of Faber polynomials.

20. John Ryan, University of Arkansas.

ABSTRACT: Using Dirac operators in n-dimensional Euclidean space one can construct analogues of the Cauchy integral formula and Plemelj projection operators. These operators are invariant under Moebius transformations, and as such have natural analogues in the setting of conformally flat manifolds. Conformally flat manifolds play the role of Riemann surfaces in n real dimensions. The purpose of this talk is to illustrate how basic results from the complex plane on Hardy spaces carry over to some examples of conformally flat manifolds.

21. Jani Onninen, University of Michigan.

ABSTRACT: Mappings of finite distortion can be thought of as a generalization of the concept of analytic functions. They include mappings of bounded distortion that are also called quasiregular mappings. In this talk, we focus on an essentially sharp modulus of continuity for mappings of finite distortion.

22. Thomas Bieske, University of South Florida.

ABSTRACT: In this talk, we discuss the relationship between absolute minimizers and infinite harmonic functions in the viscosity sense.

23. Petronela Radu, Carnegie Mellon University.

ABSTRACT: Over the past decade there has been significant progress in the study of existence of weak solutions to the semilinear wave equation with power-like damping and source terms. The results mainly look at source and damping terms of polynomial growth and they prove either existence or blow up in finite time of the solution, depending on range of exponents and on the sign of the nonlinearities.. We will present a local in time existence theorem that brings improvements to recent work by J.~Serrin, G.~Todorova and E.~Vitillaro, by extending the range of exponents and allowing more general nonlinearities. The equation under study has the form: \begin{equation}\tag{NLW} \left\{\begin{array}{l} u_{tt}-\Delta u + f(x,t,u) +g(x,t,u_t) = 0 \text{ a.e. in } \re^{n} \times [0,\infty);\\ u|_{{}_{t=0}} = u_{{}_0};\\ u_{t}|_{{}_{t=0}} = u_{{}_1}, \end{array} \right . \end{equation} where the functions $f$ and $g$ are polynomially bounded and $g$ is increasing in the last argument. The potential well technique due to L.~E.~Payne and D.~H.~Sattinger and a monotonicity argument due to J.~L.~Lions and W.~Strauss are the main tools used.

24. Alexander Stokolos, DePaul University Chicago.

ABSTRACT: Let f be positive function summable on the unit cube I. Denote by O(f,Q) the integral oscillation of the function on the cube Q and by A(f,Q) the integral average of the function f on the cube Q. Trivially, O(f,Q) < 2A(f,Q) for all cubes Q which are subcubes of I. Gurov-Reshetnyak Lemma states that if one replace 2 with sufficiently small number then the function f will be in L^p(I) for some p > 1. In the talk I will discus some improvement and generalizations of this phenomena.

25. Malgorzata Stawiska, Purdue University.

ABSTRACT: Discussing relations between complex analysis and differential topology, we give a proof of Riemann- Hurwitz formula using index theorems.

26. Clint Richardson, Stephen F. Austin State University

ABSTRACT: For the standard class S of normalized analytic functions f univalent on the unit disk U, strict lower bounds on the minimal area of the image f(U) concentrated in any given half-plane are found and explicit formulae given for the extremal maps. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regard minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.

The Prairie Analysis Seminar is a joint project of the Department of Mathematics of The University of Kansas and the Department of Mathematics of Kansas State University.