Related papers: Quantitative unique continuation of solutions to h…
We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=\mu+\mu^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the…
Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities…
In this paper we study unique continuation theorems for magnetic Schr\"odinger equation via Carleman estimates. We use integration by parts techniques in order to show these estimates. We consider electric and magnetic potentials with…
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable…
In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and $L^{1}$ datum in the setting of variable exponent Sobolev…
This paper investigates the quantitative weak unique continuation property (QWUCP) for a class of high-dimensional elliptic equations with interior point degeneracy. First, we establish well-posedness results in weighted function spaces.…
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the…
We prove a unique continuation result for an ill-posed characteristic problem. A model problem of this type occurs in A.D.~Ionescu \& S.~Klainerman article (Theorem 1.1 in \cite{MR2470908}) and we extend their model-result using only…
In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in }…
In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and…
Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain…
We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for…
We prove the existence and uniqueness of weak solutions to a class of anisotropic elliptic equations with coefficients of convection term belonging to some suitable Marcinkiewicz spaces. Some useful a priori estimates and regularity results…
In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these…
We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in…
In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_\gamma$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique…