arXiv subjects
Publications and source records attributed to Tadeusz Kulczycki.
We obtain sharp estimates for functions harmonic with respect to $x$-dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on the construction of global barrier functions for the $x$-dependent rectilinear fractional Laplacian in balls.
In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to $0^+$ and to $+\infty$. Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meet the convex envelope of the fixed boundary.
We study Schrödinger operators on $\mathbb{R}^2$ $$ H = \left(-\frac{\partial^2}{\partial x_1^2}\right)^{α/2} + \left(-\frac{\partial^2}{\partial x_2^2}\right)^{α/2} + V, $$ for $α\in (0,2)$ and some sufficiently regular, radial, confining potentials $V$. We obtain necessary and sufficient conditions on intrinsic ultracontractivity for semigroups $\{e^{-tH}: \, t \ge 0\}$. We also get sharp estimates of first eigenfunctions of $H$.
We study the structure of solutions of the interior Bernoulli free boundary problem for $(-Δ)^{α/2}$ on an interval $D$ with parameter $λ> 0$. In particular, we show that there exists a constant $λ_{α,D} > 0$ (called the Bernoulli constant) such that the problem has no solution for $λ\in (0,λ_{α,D})$, at least one solution for $λ= λ_{α,D}$ and at least two solutions for $λ> λ_{α,D}$. We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for $α= 1$ that there exist solutions of the interior Bernoulli free boundary problem for $(-Δ)^{α/2}$ on an interval which are not minimizers of the corresponding variational problem.
We study SDE $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \geq 0 $$ where $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric $α_i$-stable process ($α_i \in (0,2)$ and they are not necessarily equal). Under certain assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probability density and we establish H{ö}lder regularity of the corresponding transition semigroup. The method we propose is based on a reduction of an SDE with a drift term to another SDE without such a term but with coefficients depending on time variable. Such a method have the same spirit with the classic characteristic method and seems to be of independent interest.
We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational approach, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.
We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit `principal part' and a `residual part', subject to certain $L^\infty(dx)\otimes L^1(dy)$ and $L^\infty(dx)\otimes L^\infty(dy)$-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.
We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $R^d$, where $α\in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $γ\in (0,α)$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.
We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots, Z_t^{(d)}$ are independent one-dimensional L{é}vy processes with characteristic exponents $ψ_1, \ldots, ψ_d$. We assume that each $ψ_i$ satisfies a weak lower scaling condition WLSC($α,0,\underline{C}$), a weak upper scaling condition WUSC($β,1,\overline{C}$) (where $0< α\le β< 2$) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all $ψ_1, \ldots, ψ_d$ are the same and $α, β$ are arbitrary, or (ii) not all $ψ_1, \ldots, ψ_d$ are the same and $α> (2/3)β$. We also assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed $γ\in (0,α) \cap (0,1]$ the semigroup $P_t$ of the process $X$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. We also show the existence of a transition density of the process $X$.
In the present work we study solutions of the problem $-(-Δ)^{α/2}u = f(x,u)$ in $D_0\setminus \overline{D}_1$, with exterior conditions $u = 0$ in $R^N \setminus D_0$ and $u = 1$ in $\overline{D}_1$, where $D_1, D_0 \subset R^N$ are open sets such that $\overline{D}_1 \subset D_0$, $α\in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x} \in \overline{D}_1$, then the same occurs for every superlevel set of $u$.
We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and Hölder continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove Hölder and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.
Under some mild assumptions on the Levy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure-jump isotropic unimodal Levy processes in $R^d$.
Let $X$ be a symmetric, pure jump, unimodal Levy process in $\mathbb{R}$ with an infinite Levy measure. We prove that for any fixed $t > 0$ the survival probability $P^x(τ_{(-a,a)} > t)$ is nondecreasing on $(-a,0]$, nonincreasing on $[0,a)$ and concave on $(-a/2,a/2)$, where $a > 0$ and $τ_{(-a,a)}$ is the first exit time of the process $X$ from $(-a,a)$. We also show a similar statement for sets $(-a,a) \times F \subset \mathbb{R}^d$.
In the paper we numerically study positions of high spots (extrema) of the fundamental sloshing mode of liquid in an axisymmetric tank. Our approach is based on a linear model reducing the problem to appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method of making images of oscillating fluid. We also describe the relation of the high spot problem to the celebrated hot spots conjecture.
For a sufficiently regular open bounded set $D \subset R^2$ let us consider the equation $(-Δ)^{1/2} φ(x) = 1$, $x \in D$ with the Dirichlet exterior condition $φ(x) = 0$, $x \in D^c$. $φ$ is the expected value of the first exit time from $D$ of the Cauchy process in $R^2$. We prove that if $D \subset R^2$ is a convex bounded domain then $φ$ is concave on $D$. To show it we study the Hessian matrix of the harmonic extension of $φ$. The key idea of the proof is based on a deep result of Hans Lewy concerning determinants of Hessian matrices of harmonic functions.
In the paper we study the shape of the fundamental sloshing mode of a liquid in a container. Our approach is based on a mathematical model which is an appropriate Steklov eigenvalue problem. We present numerical results, a physical experiment, photos of a surface of oscillating liquid and also rigorous mathematical results. We describe the relation of the above problem with the celebrated hot spots conjecture.
The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.
We prove gradient estimates for harmonic functions with respect to a $d$-dimensional unimodal pure-jump Levy process under some mild assumptions on the density of its Levy measure. These assumptions allow for a construction of an unimodal Levy process in $\R^{d+2}$ with the same characteristic exponent as the original process. The relationship between the two processes provides a fruitful source of gradient estimates of transition densities. We also construct another process called a difference process which is very useful in the analysis of differential properties of harmonic functions.