Selected topics from "Notes on the p-Laplace equation" by P. Lindqvist
We continue discussion of basic properties for the p-Laplacian
We discuss generalization of the p-Laplace operator, the so-called A-harmonic operators. The presentation is based largely on Chapters 3 and 6 from HKM book.
The topic of my presentation will be Maximum and sweeping principles for quasilinear inequalities. I plan to state the maximum principle theorem and some remarks. Later I will come to the proof of two minor theorems and a sweeping theorem. Next if I have time I would like to present (at least state) some facts on comparison theorems for divergence structure inequalities.
In my presentation we are going to study a variable exponent Sobolev space on a Riemannian manifold. Continuous and compact embedding will be shown in the case of a compact manifold. As an application we will study a PDE problem involving the p(x)-Laplacian.
We will consider a stationary inhomogenous nonlinear system that generalizes p-laplacian over Orlicz-type growths. Basing on paper by Diening, Kaplicky and Schwarzacher ("BMO estimates for the p-Laplacian", Nonlinear Anal. 75 (2012), no. 2, 637-650) we present nonlinear Calderon-Zygmund theory in BMO and Campanato spaces for such systems.
I will try to give a self-contained account, aimed at non-experts, on the state of art of the mathematics behind geometrically defined curvature energies. (Part of the motivation comes from the desire of physicists to model self-avoidance phenomena for thin objects like strands, rods, sheets etc.) The basic building block, serving as a multipoint potential, is the circumradius of three points on a curve. The energies we study are defined as averages of negative powers of that radius over all possible triples of points along the curve (or via a mixture of averaging and maximization). For a suitable range of exponents, above the scale invariant case, we establish self-avoidance and regularizing effects and discuss various applications in geometric knot theory, as well as generalizations to surfaces and higher-dimensional submanifolds.
The talk is based on joint works with H. von der Mosel, M. Szumanska, and S. Kolasinski.
We will first show an estimate of Lebesgue norms of gradient of a Sobolev function due to Strzelecki and Riviere. The estimate involves only a fixed power (namely 2) of the function's second derivative and BMO norm of the function. The proof follows by Fefferman's duality. An argument similar to that used in the famous paper of Coifman et al. is used to show that certain expression belongs to the Hardy space. Then, we will apply this result to obtain regularity of solutions to p-parabolic systems with rhs bounded only by p-th power of gradient, provided that their BMO norm is small uniformly in time. Such systems arise as Euler-Lagrange equations associated with p-th energy functionals on functions with image constrained to a manifold.
During the talk I will present some basic properties of capacities. The relation between capacity and Hausdorff dimension will be studied. Theorems and definitions will be illustrated by examples. The talk will be based mainly on the book "Nonlinear Potential Theory of Degenerate Elliptic Equations" by Heinonen, Kilpelainen and Martio.
Carnot groups are the ideal models of Sub-Riemannian geometry which is loosely described by the saying: "Carnot groups are to Sub-Riemannian geometry as $\mathbb{R}^n$ is to Riemannian geometry". A defining feature of a Carnot group is a natural notion of dilation which together with left translation provide analogues of the fundamental operations of dilation and translation required to do analysis on Euclidean space. In this talk I will introduce these structures, their fundamental properties and some analogues of well known theories and ideas from $\mathbb{R}^n$ .
In the second talk I will discuss the rigidity problem and Tanaka prolongation.
As we seek greater knowledge about the energy-minimal deformations in Geometric Function Theory and Nonlinear Hyperelasticity, the questions about Sobolev homeomorphisms and their limits become ever more quintessential. We shall discuss the following topics:
My talk will be focused on derivating weighted Hardy inequalities in $L^p$, as well as in Orlicz setting. First, I am going to present brief summary of results linking PDEs and Hardy inequalities. Next, I am going to show main features of the technique used in the proofs. Such a technique is constructive and easy to conduct. It leads among others to the classical Hardy inequality with optimal constant.
A Carnot group $G$ is a nilpotent Lie group with a geometric structure; these arise in modelling sub-elliptic operators, nonholonomic systems, and sub-Riemannian geometry. A coordinate change, that is, a bijective map $\phi: \Omega \to G$, where $\Omega$ is an open subset of $G$, may be described geometrically as contact, quasiconformal or conformal. We show that conformal mappings are affine, except in a few special cases, and that if the group $G$ is rigid, that is, the space of contact mappings is finite-dimensional, then so are quasiconformal maps. This is joint work with Alessandro Ottazzi.
Minimizing harmonic maps (i.e. minimizers of the Dirichlet integral) with prescribed boudary conditions from the ball to the sphere may have singularities. For some boundary data it is known that all minimizers of the energy have singularities and the energy is strictly smaller than the infimum of the energy among the continuous extensions (the so called Lavrentiev gap phenomenon occurs). In the first part of the talk I will present a brief summary of the known results about the behavior of singularities. Then I will focus on the typicality of the Lavrentiev gap phenomenon, which is a work in progress in cooperatrion with P. Strzelecki.
Podana zostanie charakteryzacja funkcji rzeczywisto-analitycznych w terminach srednich calkowych. Charakteryzacja ta umozliwia zdefiniowanie funkcji analitycznych na przestrzeniach metrycznych.