Search results for "Harmonic"

$n$-harmonic coordinates and the regularity of conformal mappings

2014

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.

Conformality and $Q$-harmonicity in sub-Riemannian manifolds

2016

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.

$\varepsilon $-approximability of harmonic functions in $L^p$ implies uniform rectifiability

2019

The Obstacle Problem in a Non-Linear Potential Theory

1988

M. Brelot gave rise to the concept harmonic space when he extended classical potential theory on ℝn to an axiomatic system on a locally compact space. I have recently constructed1 a non-linear harmonic space by dropping the assumption that the sum of two harmonic functions is harmonic and considering some other axioms instead. This approach has its origin in the work of O. Martio, P. Lindqvist and S. Granlund2,3,4, who have developed a non-linear potential theory on ℝn connected with variational integrals of the type ∫ F(x,∇u(x)) dm(x), where F(x, h) ≈ |h|p.

The Factorization Method for Electrical Impedance Tomography in the Half-Space

2008

We consider the inverse problem of electrical impedance tomography in a conducting half-space, given electrostatic measurements on its boundary, i.e., a hyperplane. We first provide a rigorous weak analysis of the corresponding forward problem and then develop a numerical algorithm to solve an associated inverse problem. This inverse problem consists of the reconstruction of certain inclusions within the half-space which have a different conductivity than the background. To solve the inverse problem we employ the so-called factorization method of Kirsch, which so far has only been considered for the impedance tomography problem in bounded domains. Our analysis of the forward problem makes u…

Probing the Dynamics of a Molecular Ion with Laser Pulses

2004

The dynamics of a H2+ molecular ion driven by two laser pulses separated by a time lag is studied beyond the Born-Oppenheimer approximation. The first, short, pulse prepares the molecule in some quantum state, which is probed by the second pulse. Under suitable conditions, the molecule emits a spectrum of redshifted high-order harmonics. The value of the redshift is proportional to the harmonic order and can be used as a measure of the speed of the atoms of the molecule.

Microwave Response of Ceramic MgB2 Samples

2009

The microwave response of ceramic MgB2 has been investigated as a function of temperature and external magnetic field by two different techniques: microwave surface impedance and second-harmonic emission measurements. The measurements of the surface resistance have shown that microwave losses in MgB2 are strongly affected by the magnetic field in the whole range of temperatures below Tc, even for relatively low field values. The results have been accounted for in the framework of the Coffey and Clem model hypothesizing that in different temperature ranges the microwave current induces fluxons to move in different regimes. In particular, the results at temperatures close to Tc have been quan…

Alignement moléculaire : caractérisation et application à la mesure de thermalisation ultra-rapide et au contrôle de génération d'harmoniques

2013

The thematic of this thesis is molecular alignment. The latter is a very important topic that opens the way toward a much more thin control of many phenomenons. So, we have developed a new measurement technique of the molecular alignment along one axis that permits to preserve the sign of alignment. This one is, like other measurement techniques developed by the team,based on the measurement of the refractive index variation induced by the molecular alignment.The technique developed then also permits the molecular alignment measurement, being also an application of it because it allows the third harmonic generation. In the last study, molecular alignment is implemented to show that it bring…

Harmonic generation in all-dielectric metasurfaces

2023

Nonlinear light generation is a key phenomenon in many optical systems. Recently, the field of nonlinear optics has moved to the miniaturization of conventional bulky components. Among all the new platforms that have been proposed, dielectric nanoscale resonators represent excellent candidates for light generation and manipulation. When arranged in periodic arrays, such high refractive index scattering components become an artificial material called metasurface. Several approaches for designing platforms with enhanced optical nonlinearities at moderate pump intensities have been proposed. In this chapter, we review the most recent results on second- and third-order nonlinear processes in hi…

Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

2021

AbstractIn this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the prop…