arXiv subjects

Hajer Bahouri

Publications and source records attributed to Hajer Bahouri.

24 records · Page 2Linked to original sources

arXiv2012

Stability estimates for an inverse scattering problem at high frequencies

We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case.

math.AP↗

arXiv2011

Lack of compactness in the 2D critical Sobolev embedding, the general case

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.

math.AP↗

arXiv2011

A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(ϕ_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by Gérard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces.

math.FA↗

arXiv2010

On the lack of compactness in the 2D critical Sobolev embedding

This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.

math.AP↗

arXiv2008

The heat kernel and frequency localized functions on the Heisenberg group

The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to positive indexes using Bernstein inequalities. As a corollary we obtain a proof of refined Sobolev inequalities in W^{s,p} spaces.

math.AP↗

arXiv2003

Quasilinear wave equations and microlocal analysis

In this text, we shall give an outline of some recent results (see \ccite{bahourichemin2} \ccite{bahourichemin3} and \ccite{bahourichemin4}) of local wellposedness for two types of quasilinear wave equations for initial data less regular than what is required by the energy method. To go below the regularity prescribed by the classical theory of strictly hyperbolic equations, we have to use the particular properties of the wave equation. The result concerning the first kind of equations must be understood as a Strichartz estimate for wave operators whose coefficients are only Lipschitz while the result concerning the second type of equations is reduced to the proof of a bilinear estimate for the product of two solutions for wave operators whose coefficients are not very regular. The purpose of this talk is to emphasise the importance of ideas coming from microlocal analysis to prove such results.

math.AP↗