Related papers: Lipschitz metric for the two-component Camassa--Ho…

We extend the results of a work by L. H\"ormander in 1990 concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a…

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified…

This article is devoted to completing some aspects of the classical Cauchy-Lipschitz (or Picard-Lindel\"of) theory for general nonlinear systems posed on time scales, that are closed subsets of the set of real numbers. Partial results do…

We discuss a Lipschitz truncation technique for parabolic double-phase problems of $p$-Laplace type in order to prove energy estimates and uniqueness results for the Dirichlet problem. Moreover, we show existence for a non-homogeneous…

We consider the inverse problem of the simultaneous identification of the coefficients $\sigma$ and $q$ of the equation div$(\sigma\nabla u) + qu=0$ from the knowledge of the complete Cauchy data pairs. We assume that $\sigma=\gamma A$…

We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…

In this paper, we propose a multi-component system of Camassa-Holm equation, denoted by CH($N$,$H$) with 2N components and an arbitrary smooth function $H$. This system is shown to admit Lax pair and infinitely many conservation laws. We…

We prove a global well-posedness result for the Landau-Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any…

We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- \Delta)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$,…

We prove that for any second-order, homogeneous, $N \times N$ elliptic system $L$ with constant complex coefficients in $\mathbb{R}^n$, the Dirichlet problem in $\mathbb{R}^n_+$ with boundary data in $\mathrm{CMO}(\mathbb{R}^{n-1},…

In this study, we address the inverse problem of recovering the Lam\'e parameters ($\lambda, \mu$) and the density $\rho$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation…

We consider the problem to reconstruct a wave speed $c \in C^\infty(M)$ in a domain $M \subset \R^n$ from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map $\Lambda$. We introduce a reconstruction formula…

In this paper we prove a quantitative form of the strong unique continuation property for the Lam\'e system when the Lam\'e coefficients $\mu$ is Lipschitz and $\lambda$ is essentially bounded in dimension $n\ge 2$. This result is an…

The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$ which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This paper contains…

This paper deals with the Klein-Gordon equation on the Poincar\'e chart of the 5-dimensional Anti-de Sitter universe. When the mass $\mu$ is larger than $-{1}{4}$, the Cauchy problem is well posed despite the loss of global hyperbolicity…

The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ determines a flow of conservative solutions taking values in the space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t. the natural $H^1$ distance. Aim of this paper is to…

We consider the inverse boundary value problem of determining the potential $q$ in the equation $\Delta u + qu = 0$ in $\Omega\subset\mathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension…

We consider a 3$\times$3 spectral problem which generates four-component CH type systems. The bi-Hamiltonian structure and infinitely many conserved quantities are constructed for the associated hierarchy. Some possible reductions are also…

We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting…