Related papers: \alpha-Continuity Properties of Stable Processes

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$ and let $d_\Gamma$ denote the Euclidean distance to $\Gamma$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous…

Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which…

In a series of papers, M.Talagrand, the second author and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

In this paper we consider the specification property for $(\alpha,\beta)$-shifts. When $\alpha=0$, Schmeling shows that the set of $\beta>1$ for which the $\beta$-shift has the specification property has the Lebesgue measure zero but has…

We employ a nonlocal method to study the asymptotic behavior at infinity ofsolutions to the two-dimensional supercritical Lagrangian mean curvature equation \[ \arctan \lambda_1(D^2u)+\arctan \lambda_2(D^2u) = \theta + f(x) \] on exterior…

We prove the intrinsic ultracontractivity for the semigroup generated by a large class of symmetric Levy processes such that the Levy measure satisfies some conditions in the neighborhood of 0, killed on exiting a bounded and connected…

We consider Laplacian eigenfunctions on a domain $\Omega \subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary…

Let $ \overline B=\{ \overline B_{t},t\in R^{1} \}$ be Brownian motion killed after an independent exponential time with mean $2/\lambda^{2}$. The process $\overline B$ has potential densities, \[ u(x,y) ={e^{-\lambda |y-x|}\over…

An identity in law for the area of a spectrally positive L\'evy stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse Beta random variable and the square of a positive stable…

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ which vanish in a suitable trace sense on…

In classical analysis, Lebesgue first proved that $\mathbb{R}$ has the property that each Riemann integrable function from $[a,b]$ into $\mathbb{R}$ is continuous almost everywhere. This property is named as the Lebesgue property. Though…

For $0<\alpha\leq2$, a super-$\alpha$-stable motion $X$ in $\mathsf{R}^d$ with branching of index $1+\beta\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<\alpha/\beta$, a dichotomy for the density function of the measure $X_t$ holds:…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

Let $X=\{X(t),t\in R_+\}$ be a real-valued symmetric L\'{e}vy process with continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{i\lambda X(t)}=e^{-t\psi(\lambda)}$. Let…

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open…

The Laplacian $\Delta_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $\Lambda$, the Dirichlet-to-Neumann map of the unit ball, as…

We show that for $0<\gamma, \gamma' <1$ and for measurable subsets of the unit square with Lebesgue measure $\gamma$ there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the…

We consider the following eigenvalue optimization problem: Given a bounded domain $\Omega\subset\R^n$ and numbers $\alpha\geq 0$, $A\in [0,|\Omega|]$, find a subset $D\subset\Omega$ of area $A$ for which the first Dirichlet eigenvalue of…