Startsida
Earlier seminars, 2011
Abstract: Pseudodifferential operators in $R^n$
Abstract: We shall discuss properties of the discrete and continuous spectrum of different classes of self-adjoint differential operators including Schrödinger operators.
Abstract: One of the main breakthroughs in the 20th century mathematics was De Giorgi's proof of Hölder continuity for solutions of elliptic PDEs. His method has since then been used to prove interior regularity in various contexts. It is maybe less known, though not entirely surprising that De Giorgi's method also yields sufficient conditions for boundary regularity. In the talk, I will discuss a recent variation of De Giorgi's method which goes in the opposite direction, leading to a necessary condition for boundary regularity of PDEs.
Abstract: Sobolev spaces and pseudodifferential operators in $R^n$. (The "introduction" last time was probably hard to follow. This seminar will be much more elementary.)
Abstract: I will talk about joint work with Tuomas Hytönen. Our preprint, available at arXiv, has the following abstract. As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L_2-Whitney averages belongs to L_2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and Rosén, we find the pre-dual of X, and characterize the pointwise multipliers from X to L_2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L_p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.
Abstract: The index theorem of M.F. Atiyah and I.M. Singer, first announced in 1963, is a deep result in mathematics, providing a fundamental link between analysis and topology. This series of seminars aims at giving a reasonably elementary and complete presentation of the mathematics involved in the statement and proof of the K-theoretical version of the index theorem.
Abstract: We will start by reviewing the origins of the classical Hardy's inequality (1920), for infinite series or integrals, finding the best constants in the power weight case, and study some recent developments in the theory of weighted inequalities for the Hardy operator, with applications to normability properties of Function Spaces.
Secondly, we give a positive answer to a conjecture of N. Kruglyak and E. Setterqvist (2008) about the norm of the Hardy operator minus the identity on decreasing functions, extending their result to the full range of indices and to a broader class of weights (joint work with S. Boza).
Abstract: The talk concerns the first boundary value problem for the heat equation (and more general parabolic equations) in a cone or dihedron. The asymptotics of the solutions near the vertex of the cone and the edge of the dihedron is studied. Here the remainder belongs to a weighted $L_p$ Sobolev space, where the weight is a power of the distance from the vertex/edge.
Abstract: The first part of the present talk concerns the problem describing the time-harmonic small-amplitude motion of the mechanical system that consists of a three-dimensional water layer and a body (either surface-piercing or totally submerged), freely floating in it. The latter means that there are no external forces acting on the body, for example, due to constraints on the body motion. Water extends to infinity in horizontal directions, but has a finite depth being bounded from below by a horizontal rigid bottom and from above by a free surface. Thus our model describes a vessel freely floating in an open sea of a constant depth. The coupled boundary value problem under consideration contains a spectral parameter -- the frequency of oscillations -- in the boundary conditions as well as in the equations governing the body motion. We prove that the total energy of the water motion is finite and establish the equipartition of energy of the whole system. Under certain restrictions on body's geometry the problem is proved to have only a trivial solution for sufficiently large values of the frequency. The uniqueness frequencies are estimated from below.
In the second part of the talk, water occupies a half-space, whereas the body is again either surface-piercing or totally submerged. Again, it is proved that the total energy of the water motion is finite, the equipartition of energy holds for the whole system, and under certain restrictions on the body geometry the problem has only a trivial solution for sufficiently large frequencies. Besides, infinitely many eigensolutions is constructed for any frequency; for this purpose the so-called semi-inverse procedure is applied. Each of these modes is trapped by infinitely many bodies (this is proved rigorously for some classes of bodies). The trapping bodies are surface-piercing, have axisymmetric submerged parts, that violate the geometric conditions of the corresponding uniqueness theorem; all of the bodies are motionless although float freely.
Abstract: In the vast subject of weighted inequalities, we consider the question of whether or not the maximal function maps $L^p(\sigma)$ into $L^p(w)$, $\sigma$ and $w$ being two positive Borel measures on $\mathbb{R}^d$. With the help of the weighted maximal function, we prove Sawyer's two weight maximal function theorem, which characterizes the two weight inequalities for the maximal function.
Abstract: This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An e¤ective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end a general class of problems is studied that, e.g., includes the incompressible Reynolds problem in both cartesian and cylindrical coordinate forms. To demonstrate the e¤ectiveness of our method, several numerical results are presented that clearly show the convergence of the deterministic solutions toward the homogenized solution. Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.
Abstract: F. John has formulated in 1949 the spectral boundary value problem on a freely floating object. For more than 50 years the research has been focusing on the case of fixed obstacles and many remarkable results have been obtained within this topic. A new reduction scheme is presented which reduces a qudratic pencil, that is an abstract realization of the John problem on a freely floating object, to standard spectral problem for a bounded self-adjoint operator. Many new results can be derived by using this method, some of them related to trapped modes and localization estimates for them, will be discussed.
Abstract: The boundaries of quadrature domains are a type of free boundaries, and it is important to understand how they behave. Roughly speaking one can say that they are real analytic apart from certain cusp-like singularities. The theory of free boundary regularity by now has a standard machinery which to a large extent is adapted from corresponding material from geometric measure theory. In this talk we will look at the simplest case which is the classical obstacle problem, and sketch the proof of the fact that at points of the free boundary where the complement is big (in this case satisfies an exterior cone condition) the boundary is locally real analytic.
Abstract: This seminar will introduce the three basic types of quadrature domains, i.e. those for analytic, harmonic and subharmonic functions. We will discuss their basic properties, what types of questions that one usually looks at and try to motivate why they deserve to be studied.
Abstract: This is the first seminar in a series of three which will concern quadrature domains and related topics. None of the seminars will really be about my own research results, but is rather intended to be an accessible introduction to the area. The first of these lectures will be a crash-course in linear potential theory for the Laplace operator. We will discuss basic properties of harmonic and subharmonic functions and the solution of Dirichlet's problem by the so called Perron method.
Abstract:Current studies on analysis in metric measure spaces focus on a $p$-Poincaré inequality for finite values of $p$. For $p=1$ and $p=Q$ (when the measure is Ahlfors $Q$-regular) these are known to be geometric in nature. Because of the Hölder inequality, the weakest possible Poincaré inequality corresponds to $p=\infty$.
Abstract: We give a review of results concerning the multiplicity of positive solutions of elliptic BVPs for quasilinear equations of variational structure which is extensively studied during last two decades. Namely, for some BVPs we demonstrate how to obtain arbitrarily many nonequivalent positive solutions by choosing parameters in a proper way. |
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Last updated: 2011-12-21
