Related papers: Quantitative unique continuation on conic manifold…
We prove that compact topological 4-manifolds can be effectively presented by a finite amount of data.
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for…
This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.
This article deals with the weak and strong unique continuation principle for fractional Schr\"odinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable…
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility…
We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then…
We prove the strong unique continuation property for many-body Pauli operators with external potentials, interaction potentials and magnetic fields in $L^p\loc(\R^d)$, and with magnetic potentials in ${L^{q}\loc(\R^d)}$, where ${p >…
We give a sharp upper bound on the vanishing order of solutions to Schrodinger equation with C^1 electric and magnetic potentials on a compact smooth manifold. Our method is based on quantitative Carleman type inequalities developed by…
A simplified, user-friendly repackaging of the curvature estimates implied by the Seiberg-Witten equations is formulated in terms of the convex hull of the set of monopole classes. New results are also obtained concerning boundary cases of…
We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.
We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schr\"odinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting…
This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding…
This is the sequel to our first paper concerning the balanced embedding of a non-compact complex manifold into an infinite-dimensional projective space. We prove the uniqueness of such an embedding. The proof relies on fine estimates of the…
We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$…
We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schr\"odinger operators on Riemannian manifolds with Ricci curvature bounded below. The upper bound for the…
In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group $\mathbb G$ for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp…
This article develops the numerical and theoretical study of a reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate.…
In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.
We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary…