Related papers: On generalized Holmgren's principle to the Lam\'e …

There are numerous studies focusing on the convergence of the principal eigenvalue $\lambda(s)$ as $s\to+\infty$ corresponding to the elliptic eigenvalue problem \begin{align*}…

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity…

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int…

The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lam\'{e} system with only bounded and measurable coefficients. Second, we derive a…

We consider the highly nonlinear and ill-posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu-\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$, with $T>0$…

Let $M$ be a Riemannian manifold, $\tau: G \times M \to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \to \mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\"odinger operator, $P=-\hbar^2…

This paper is devoted to studying the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta<0,\gamma>0$. Firstly, by using the…

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = \psi(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically…

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a nonlocal operator.…

We examine the equation \[\Delta^2 u = \lambda f(u) \qquad \Omega, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for…

We consider abstract nonlinear equations of the form $A u = \lambda u + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $\lambda \in \mathbb{R}$ is a parameter, and $u \mapsto I'(u)$ is a…

We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\…

We consider generic properties of Lagrangians. Our main result is the Theorem of Kupka-Smale, in the Lagrangian setting, claiming that, for a convex and superlinear Lagrangian defined in a compact surface, for each $k\in \mathbb{R}$,…

Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$ for mappings $u:M\to N$, i.e., $L$ is an expression involving $u$ and its first derivatives whose value is an $m$-form (or more generally, an $m$-density) on~$M$.…

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 $\gamma=\lambda - 1$…

In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem \begin{equation} \left\{ \begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0…

Let $\Omega$ be a bounded $C^{2}$ domain in $\R^n$, and let $\Omega^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\div(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with…

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the…

We consider the inverse obstacle scattering problem of determining both the shape and the "equivalent impedance" from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface…

In this paper, we develop a general approach to prove stability for the non linear second step of hybrid inverse problems. We work with general functionals of the form $\sigma|\nabla u|^p$, $0 < p \leq 1$, where $u$ is the solution of the…