Tadeusz Kulczycki

Estimates and structure of α-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on Dc∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.

Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes

Let X t be the relativistic α-stable process in R d , a ∈ (0, 2), d > α, with infinitesimal generator H (α) 0 = -((-A + m 2/α ) α/2 - m). We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup T t for this process with generator H (α) 0 - V, V > 0, V locally bounded. We prove that if lim |x|→∞ V(x) = ∞, then for every t > 0 the operator T t is compact. We consider the class V of potentials V such that V > 0, lim |x|→∞ V(x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup T t is IU if and only if lim |x|→∞ V(x)/|x| = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction Φ 1 for T t . In particular, when V(x) = |x| β , β > 0, then the semigroup T t is IU if and only if β > 1. For β > 1 the first eigenfunction Φ 1 (x) is comparable to.

Spectral properties of the Cauchy process on half-line and interval

We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0, ∞ )a nd the interval ( −1, 1). This process is related to the square root of one-dimensional Laplacian A = − � −(d 2 /dx 2 ) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions ψλ of A is derived, and then used to construct a spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process on the half-line (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ/2 − π/ 8+ O(1/n) is derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point.

On the Shape of the Ground State Eigenfunction for Stable Processes

Abstract We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (−1,1) is concave on

Spectral gap for the Cauchy process on convex, symmetric domains

(-\frac{1}{2},\frac{1}{2})

On the 'high spots' of fundamental sloshing modes in a trough

. We call this property “mid-concavity”. A similar statement holds for rectangles in ℝd, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination.

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

Let D ⊂ ℝ2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b > 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λ n } n=1 ∞ be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap: where C is an absolute constant. The estimate is obtained by proving new weighted Poincaré inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem established in Bañuelos and Kulczycki (2004).

The shape of the fundamental sloshing mode in axisymmetric containers

We study an eigenvalue problem with a spectral parameter in a boundary condition. The problem describes sloshing of a heavy liquid in a container, which means that the unknowns are the frequencies and modes of the liquid’s free oscillations. The question of ‘high spots’ (the points on the mean free surface, where its elevation attains the maximum and minimum values) is considered for fundamental sloshing modes in troughs of uniform cross section. For troughs, whose cross sections are such that the horizontal, top interval is the one-to-one orthogonal projection of the bottom, the following result is obtained: any fundamental eigenfunction attains its maximum and minimum values only on the boundary of the rectangular free surface of the trough.

Stationary Distributions for Jump Processes with Inert Drift

We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W in R^3. We study the case when W is an axially symmetric, convex, bounded domain satisfying the John condition. The Cartesian coordinates (x,y,z) are chosen so that the mean free surface of the liquid lies in (x,z)-plane and y-axis is directed upwards (y-axis is the axis of symmetry). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction phi there is a change of x,z coordinates by a rotation around y-axis so that phi is odd in x-variable. The second result of the paper gives the following monotonicity property of the fundamental eigenfunction phi. If phi is odd in x-variable then it is strictly monotonic in x-variable. This property has the following hydrodynamical meaning. If liquid oscillates freely with fundamental frequency according to phi then the free surface elevation of liquid is increasing along each line parallel to x-axis during one period of time and decreasing during the other half period. The proof of the second result is based on the method developed by D. Jerison and N. Nadirashvili for the hot spots problem for Neumann Laplacian.

Properties of Green function of symmetric stable processes

We numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model and reduces the problem to an appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method for making images of an oscillating fluid.