Related papers: On the First Eigenfunction of the Symmetric Stable…

Let $P$ be the isotropic nearest neighbor transition operator on a homogeneous tree. We consider the $\lambda$-eigenfunctions of $P$ for $\lambda$ outside its $\ell^2$ spectrum, i.e., the eigenfunctions with eigenvalue $\gamma=\lambda - 1$…

We investigate the following eigenvalue problem \begin{align*} \begin{cases} -\operatorname{div}\left( L(x) |\nabla u| ^{p-2}\nabla u\right)=\lambda K(x)|u|^{p-2}u \quad \text{in } A_{R_1}^{R_2} , u=0\quad \text{on } \partial A_{R_1}^{R_2}…

We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g.,…

We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. For dimensions $3 \leq m \leq 6$, we generalize the work of…

In this paper we consider Harnack inequalities with respect to a symmetric $\alpha$-stable L\'evy process $X$ in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We study the example from the article \cite{bg-sz-1}. There, the authors have…

We consider the open unit disk $\mathbb{D}$ equipped with the hyperbolic metric and the associated hyperbolic Laplacian $\mathfrak{L}$. For $\lambda \in \mathbb{C}$ and $n \in \mathbb{N}$, a $\lambda$-polyharmonic function of order $n$ is a…

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative,…

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq…

This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain $\Omega \subset \mathbb R^d, d\geq 2$ involving some families of positive…

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following:…

The L2-approximation of occupation and local times of a symmetric $\alpha$-stable L{\'e}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when…

We consider the solution of the problem $$ -\Delta u=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ \Omega, \ \ u=0 \ \mbox{ on } \ \Gamma, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with boundary $\Gamma$ of class $C^{2,\tau}$,…

Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schr\"{o}dinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number…

We consider the inverse problem of determining the Lam\'{e} parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse…

Let $\Omega \subset \mathbb{R}^N$, $N \ge 2$, be a bounded domain with Lipschitz boundary, divided by a Lipschitz hypersurface $\Sigma$ into two open, disjoint Lipschitz subdomains $\Omega_1$ and $\Omega_2$. We study a nonlinear…

Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space…

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…

We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose…