Related papers: On the strong unique continuation property for the…

We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace…

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…

We determine the largest non-trivial rate of exponential decay at infinity for solutions to the Dirac equation \begin{equation*} \mathcal{D}_n \psi + \mathbb{V} \psi = 0 \quad \text{ in }\mathbb{R}^n, \end{equation*} being $\mathcal{D}_n$…

We prove quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle…

We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The Dirac…

We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists…

In this work, we consider Dirac-type operators with a constant delay less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied.…

In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum…

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $\delta$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a…

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-\alpha}$ for $\alpha>0$. We recover all the spectral regimes…

We shall investigate the asymptotic behavior of the extended resolvent R(s) of the Dirac operator as |s| increases to infinity, where s is a real parameter. It will be shown that the norm of R(s), as a bounded operator between two weighted…

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove…

We consider static U(1) monopole in non-commutative space. Up to the second order in the non-commutativity scale $\theta$, we find no non-trivial corrections to the Dirac solution, the monopole mass remains infinite. We argue the same holds…

We derive a consistency result, in the $L_1$-sense, for incomplete U-statistics in the non-standard case where the kernel at hand has infinite second-order moments. Assuming that the kernel has finite moments of order $p(\geq 1)$, we obtain…

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of…

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular…

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed…

In this note, we study the complex constant rank condition for differential operators and its implications for coercive differential inequalities. These are inequalities of the form \[ \Vert A u \Vert_{L^p} \leq \Vert \mathscr{A} u…

We prove some Hardy-Dirac inequalities with two different weights including measure valued and Coulombic ones. Those inequalities are used to construct distinguished self-adjoint extensions of Dirac operators for a class of diagonal…

We revisit \cite[Theorem 6.3]{JK}. Following the main ideas used to prove this theorem, we establish a quantitative version of the strong unique continuation property for the Sch\"odinger operator with unbounded potential. We also show that…