Search results for "IAA"
showing 10 items of 3445 documents
Short time existence of the classical solution to the fractional mean curvature flow
2019
Abstract We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C 1 , 1 -regular. We provide the same result also for the volume preserving fractional mean curvature flow.
Sub-Finsler Geodesics on the Cartan Group
2018
This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces.
Non-parametric mean curvature flow with prescribed contact angle in Riemannian products
2020
Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $\Omega$ and Ricci curvature in $\Omega$.
A Cornucopia of Carnot groups in Low Dimensions
2022
Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…
Conformal equivalence of visual metrics in pseudoconvex domains
2017
We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.
Inverse problems for elliptic equations with power type nonlinearities
2021
We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calder\'on problem for certain semilinear equations in a surprisingly simple way w…
RECOVERY OF THE SOUND SPEED FOR THE ACOUSTIC WAVE EQUATION FROM PHASELESS MEASUREMENTS
2018
We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of conformal factors. We obtain new stability results for the Hamiltonian flow transform, which allow to recover the higher order terms.
Approximation by mappings with singular Hessian minors
2018
Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,\alpha}$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.
Polynomial and horizontally polynomial functions on Lie groups
2022
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous defini…
Infinitesimal Hilbertianity of Weighted Riemannian Manifolds
2018
AbstractThe main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the…