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Yonggeun Cho; Gyeongha Hwang; Hichem Hajaiej; Tohru Ozawa
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We show the existence of ground state and orbital stability of standing waves of fractional Schr\"{o}dinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
Source: http://arxiv.org/abs/1302.2719v2
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Huyuan Chen; Hichem Hajaiej; Ying Wang
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Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}$ a measure $$\langle\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha},f\rangle=\int_{\partial\Omega}\frac{\partial^\alpha f(x)}{\partial \vec{n}_x^\alpha} d\omega(x),\quad f\in C^1(\bar\Omega),$$ where $\vec{n}_x$ is the unit outward normal vector at...
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1505.02490
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Hichem Hajaiej; Marco Squassina
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We generalize PolyaSzego inequality to integrands depending on $u$ and its gradient. Under minimal additional assumptions, we establish equality cases in this generalized inequality. We also give relevant applications of our study to a class of quasilinear elliptic equations and systems.
Source: http://arxiv.org/abs/0903.3975v7
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For a large class of supermodular integrands, we establish conditions under which balls are the unique (up to translations) maximizers of the Riesztype functionals with constraints.
Source: http://arxiv.org/abs/0903.2821v1
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Huyuan Chen; Suad Alhomedan; Hichem Hajaiej; Peter Markowich
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In this paper, we classify the fundamental solutions for a class of Schrodinger operators.
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1703.04053
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Almut Burchard; Hichem Hajaiej
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The rearrangement inequalities of HardyLittlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended HardyLittlewood and...
Source: http://arxiv.org/abs/math/0506336v2
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Wael Abdelhedi; Obaid Algahtani; Hichem Chtioui; Hichem Hajaiej
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In this paper, we establish compactness and existence results to a BransonPaneitz type problem on a bounded domain of R^n with Navier boundary condition.
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1703.03878
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Hichem Hajaiej; Giovanni Molica Bisci; Luca Vilasi
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We are concerned with existence results for a critical problem of BrezisNirenberg Type involving an integrodifferential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountainpass theorem
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1607.00458
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TaiChia Lin; Milivoj R. Belic; Milan S. Petrovic; Hichem Hajaiej; Goong Chen
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The virial theorem is a nice property for the linear Schrodinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrodinger equation with powerlaw nonlinearity, then a similar ratio can be obtained but there seems no way of getting any eigenvalue estimate. It is surprising as far as we are concerned that when the...
Topics: Nonlinear Sciences, Analysis of PDEs, Physics, Mathematics, Pattern Formation and Solitons, Optics,...
Source: http://arxiv.org/abs/1608.06102
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Hichem Hajaiej; Luc Molinet; Tohru Ozawa; Baoxiang Wang
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Sufficient and necessary conditions for the generalized GagliardoNirenberg (GN) inequality in Besov spaces and TriebelLizorkin spaces are obtained. Applying the GN inequality, we show that the finitetime blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.
Source: http://arxiv.org/abs/1004.4287v3
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We prove that supermodularity is a necessary condition for the generalized Hardy Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz{type integral.
Source: http://arxiv.org/abs/1003.3166v1
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We prove the existence of radial and radially decreasing ground states of an mcoupled nonlinear Schrodinger equation with a general nonlinearity.
Source: http://arxiv.org/abs/0903.2854v1
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We establish general assumptions under which a constrained vari ational problem involving the fractional gradient and a local nonlin earity admits minimizers.
Source: http://arxiv.org/abs/1107.0396v2
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Said El Manouni; Hichem Hajaiej; Patrick Winkert
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The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of...
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1603.03973
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Vincenzo Ambrosio; Hichem Hajaiej
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This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N> 2s$, $(\Delta)^{s}$ is the fractional Laplacian, $k$ is a bounded positive function, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ is nonnegative and $f$ is either asymptotically linear or...
Topics: Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1612.02400
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Hichem Hajaiej; Stefan Krömer
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We prove a weakstrong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended PolyaSzeg\"o inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.
Source: http://arxiv.org/abs/1008.1939v1
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Yonggeun Cho; Gyeongha Hwang; Hichem Hajaiej; Tohru Ozawa
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We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = \lambda (\frac{\psi(\cdot)}{\cdot^\gamma} * u^2)u$$ with $\lambda = \pm1, 0
Source: http://arxiv.org/abs/1209.5899v2
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We prove the uniqueness of the maximizers of a HardyLittlewood type functional under constraints. We also establish a quantitative stability estimate. Introduction
Source: http://arxiv.org/abs/0903.2826v1
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Huyuan Chen; Hichem Hajaiej
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The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{(\Delta)^\alpha +\epsilon g(u)} u=0\quad &{\rm in}\quad\ \ \bar\Omega^c, \end{array} \end{equation} where $k>0$, $\epsilon=1$ or $1$, $(\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian defined...
Topics: Mathematics, Analysis of PDEs
Source: http://arxiv.org/abs/1410.2672
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We study wellposedness, local and global, existence of solutions for a general class of physically meaningful nonlinear Schr\"odinger systems with magnetic field involving local and nonlocal nonlinearities.
Source: http://arxiv.org/abs/1004.3242v2
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In this paper we prove the Fractional GagliardoNirenberg Inequality, PolyaSzego Inequality and the Sharp Fractional Sobolev Inequality, we then provide an application of such inequalities in a constraiend variational problem involving the fractional gradient and alocal nonlinearity
Source: http://arxiv.org/abs/1104.1414v1
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Rémi Carles; Hichem Hajaiej
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We study the stability of the standing wave solutions of a GrossPitaevskii equation describing BoseEinstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.
Topics: Mathematics, Analysis of PDEs, Mathematical Physics
Source: http://arxiv.org/abs/1402.7267
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In this second part, we establish the existence of special solutions of the nonlinear Schr\"odinger system studied in the first part when the diamagnetic field is nul. We also prove some symmetry properties of these ground states solutions.
Source: http://arxiv.org/abs/1004.3244v2
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Paolo Antonelli; Agisillaos Athanassoulis; Hichem Hajaiej; Peter Markowich
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We analyse a nonlinear Schr\"odinger equation for the timeevolution of the wave function of an electron beam, interacting selfconsistently through a HartreeFock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an Xray Free Electron Laser (XFEL). We prove...
Source: http://arxiv.org/abs/1209.6089v1
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Huyuan Chen; Hichem Hajaiej; Ying Wang
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The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (\Delta)^{\alpha} u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(\Delta)^{\alpha} +u^p} u=\delta_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (\Delta)^{\alpha}$ with $\alpha\in(0,1)$ denotes the fractional Laplacian operator in the principle value...
Topics: Mathematics, Analysis of PDEs
Source: http://arxiv.org/abs/1501.06242