The Bergman kernel has become a very important tool in geometric function theory, both in one and several complex variables. It turned out that not only classical Bergman kernel, but also weighted one can be useful, particularly from quantum theory point of view. Regular and weighted Bergman space will be introduced. Some applications will be shown.
I will discuss embedding results in the context of constant exponent ("classic") Lebesgue spaces, their generalization variable exponent Lebesgue spaces, and their generalizations, Musielak-Orlicz spaces and metric measure spaces. I will try to present the idea of variable exponent Lebesgue spaces through the similarities and differences of the embedding results and their proofs.
We found an identity which yields a minimum principle for the Jacobian of planar harmonic mappings, with explicit bounds in terms of its boundary values. This work leads to a new proof of the classical Rado-Kneser-Choquet Theorem. The ideas in the present talk apply to more general nonlinear PDEs for energy-minimal mappings. The talk is based on a joint work with Jani Onninen.
I will discuss embedding results for Sobolev spaces defined on LCA groups. Moreover, in the case of compact abelian groups the Rellich-Kondrachov type theorem will be presented. Also, I will state some results for spaces defined on measurable subsets of LCA groups.
The goal of the talk is to show the Trudinger-Wang method of proving the Sobolev-type inequalities for the Hessian equation and the associated m-convex functions. The proof essentially relies on a nonlinear version of the p-Laplace operator. Further motivation for our studies comes from the fact, that the analogous results for the complex Hessian equation do not hold or remain open questions.
This talk will introduce the affine dimension of a set in Euclidean space. Affine dimension is a quantity similar to Hausdorff dimension which arises in the theory of convolution operators and Fourier restriction problems.
Recently, Iwaniec and Onninen gave a new proof of the classical Rado-Kneser-Choquet theorem in the plane. The proof was based on the fact that the logarithm of the Jacobian determinant of a harmonic function is superharmonic, assuming that the Jacobian is positive. New computations have shown that even for solutions of more general Euler-Lagrange equations, there exist nonlinear differential expressions which are subharmonic (or superharmonic). This has yielded new knowledge of the solutions, for example, a generalization of the Rado-Kneser-Choquet theorem for p-harmonic mappings in the plane. We aim to give exposition to these results and classify the energy functionals which give rise to such differential expressions.
The homeomorphic solutions to a nonlinear Beltrami equation (with the ellipticity bounded by the square root of 2 near the infinity) generate a two-dimensional submanifold of the local Sobolev space of differentiability 1 and integrability 2. We will discuss quasiconformal maps and sketch the proof that uses normal family arguments and the uniqueness of solutions to the Beltrami equation.
We consider the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincare inequality. We focus on two versions of the Cartan property and their applications to the non-linear potential theory. The talk is based on two joint papers with Anders and Jana Bjorn.
It is well known that in general on noncompact spaces the Rellich-Kondrachov type theorem does not hold. In my talk I will focus on this issue.
Let $G \subsetneq \mathbb{R}^n$ be a domain and let $d_1$ and $d_2$ be two metrics on $G$. We compare the geometries defined by the two metrics to each other for several pairs of metrics. The metrics we study include the distance ratio metric, the triangular ratio metric, the visual angle metric, and the Cassinian metric. Also we study inclusion properties of metric balls say in $\mathbb{R}^n \setminus \{0\}$. This talk is based on [chkv], [hvz], [hkv1], [hkv2].
ReferenceMetrics have an important role in geometric function theory, such as the hyperbolic metric, the chordal metric and of course the euclidean metric. There are also several other metrics, to some extent similar to the hyperbolic metric, called hyperbolic type metrics. A survey of hyperbolic type metrics from the point of view of quasiconformal maps is given. Some of these metrics are the quasihyperbolic metric, the visual angle metric, the triangular ratio metric and the distance ratio metric. Also some topics dealing with the case of K-quasiconformal maps when the dilatation K is close to 1 are discussed. This talk is a partial survey of the work of the author with several coauthors. References
In this talk we will look at some of the details in the paper "Diffeomorphic Approximation of Sobolev Homeomorphisms" by T. Iwaniec, L. Kovalev and J. Onninen, Arch. Ration. Mech. Anal., 201 (2011).
Continuation of the talk from 5th of March.
I will talk on a recent manuscript with Tomasz Adamowicz, available on ArXiv , where we show the H\"older continuity of quasiminimizers of the energy functional $\int_\Omega f(x,u,\nabla u)\,dx$ with nonstandard growth under the general structure conditions
|z|^{p(x)} - b(x)|y|^{r(x)} - g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{r(x)} + g(x).
The result is illustrated by showing that weak solutions to a class of $(A,B)$-harmonic equations-\textrm{div} A(x,u,\nabla u) = B(x,u,\nabla u),
are quasiminimizers of the variational integral of the above type and, thus, are H\"older continuous. The results extend work by Chiad\`o Piat--Coscia, Fan--Zhao and Giusti--Giaquinta.We present an approximation result for Barenblatt's cohesive fracture energies in the case of antiplane shear. The regularizing functionals are damage energies of Ambrosio--Tortorelli type and the approximation is obtained in the sense of Gamma-convergence. We also discuss how the phase field convergence scheme can be applied to approximate different special fracture models, like Griffith's model, Dugdale's model, and models with surface energy density having a power-law growth at small openings. The extension to the general case of linearized elasticity in dimension n is still an open problem. We will present some theoretical tools which one has to deal with for that sake (joint works with S. Conti and M. Focardi).
We continue our research on Sobolev spaces on locally compact abelian groups. It is interesting to study these spaces when the dual group and/or the group is metrizable. For such groups we define Holder spaces and show embedding of our spaces into them. Next we discuss the Moser-Trudinger type inequalities and the trace theorem.
The Harmonic Mapping Problem asks when there exists a harmonic homeomorphism between two given domains. It is well know that such a mapping exists in the case of simply connected domains in the complex plane except the cases when one of the domains is the complex plane and the other is a proper subset. We will investigate this problem for doubly connected domains in the plane, where it already presents considerable challenge. The talk is based on the paper by T. Iwaniec, L. Kovalev and J. Onninen: "The harmonic mapping problem and affine capacity", Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 5, 1017-1030.
We shall investigate properties of null and free Lagrangians and explain how to use them in the theory of elastic deformations of domains. The talk is based on an article by T.Iwaniec and J.Onninen "An invitation to n-harmonic hyperelasticity", Pure Appl. Math. Q. 7(2) (2011).
In the talk there will be given a short review of holomorphic extension problems starting with the famous Hartogs theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent results on global holomorphic extensions for unbounded domains in C^n. The classical Hartogs theorem solves the extension problem for bounded domains in C^n and clearly shows the difference between one and several-variables cases. Surprisingly, not many results are in the unbounded case even though such kind of theorems are important not only in Complex Analysis, but also in other areas of mathematics. The talk will be illustrated by many figures and pictures. This is a common work with Al Boggess (Arizona State University) and Zbigniew Slodkowski (University of Illinois at Chicago).
A Besicovitch set is a Borel, Lebesgue-null set which contains a unit segment in each direction. Kakeya conjecture states that every Besicovitch set in R^n has the Hausdorff dimension n. The conjecture has been confirmed for n=2. During the talk I will present connections of Kakeya conjecture with well-known problems of Fourier Analysis. If time permits, I will show a lower estimate for the Hausdorff dimension of Besicovitch sets in R^n.
The talk is based on several chapters of the book "Lectures on Harmonic Analysis" by T. Wolff.