Gagliardo Nirenberg Inequality Research Articles

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Singular limit of periodic metric grids

We investigate the asymptotic behavior of nonlinear Schrödinger ground states on d-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise–affine extensions of such states converge strongly in H1(Rd) to the corresponding ground states on Rd. As an application of such convergence results, qualitative properties of ground states and multiplicity results for fixed mass critical points of the energy on grids are derived. Moreover, we compare optimal constants in d-dimensional Gagliardo–Nirenberg inequalities on Rd and on grids. For L2-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on Rd but, contrary to Rd, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo–Nirenberg inequalities on grids.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Concentration of blow-up solutions for the Gross-Pitaveskii equation

Abstract We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 {L}^{2} supercritical case. Finally, in the L 2 {L}^{2} critical case, a delicate limit of blow-up solutions is analyzed.

Sobolev trace-type inequalities via time-space fractional heat equations

Abstract This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$ . Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$ logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$ . As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials

We present a generalized version of a Gagliardo–Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behavior. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of a local or nonlocal nature.

Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem

Abstract The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.

Regularity criterion for 3D generalized Newtonian fluids in BMO

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 53<p<115 provided that the velocity field belongs to the critical space L2p−1(0,T;BMO(R3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO-norm. In particular, when 2<p<115, we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖u‖BMO and ‖|Du|p−22∇2u‖2.

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

On the discrete Sobolev inequalities

Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d  for  d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $\mathbb{R}^{d} \text { for } d \in \mathbb{N}^{*}, p \in[1,+\infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d  into  L d d − 1 $B V\left(\mathbb{R}^{d}\right) \text { into } \mathrm{L}^{\frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p &gt; d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p &gt; d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.

Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term

Abstract A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^\infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.

Stein-Weiss-Adams inequality on Morrey spaces

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on stratified groups. While the results are obtained in the setting of general homogeneous groups, they are new already for the Euclidean space RN.

Ground states for the NLS equation with combined local nonlinearities on noncompact metric graphs

We investigate the existence of ground states for the nonlinear Schrödinger equation with combined local L2-critical and subcritical nonlinearities on noncompact metric graphs. The interplay of the combined local nonlinearities creates some new phenomena. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on the topological and metric properties of the noncompact metric graphs. Our results rely on the Concentration-compactness principle and an application of Gagliardo-Nirenberg inequalities.

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation i∂tu + Δu = −|x|b|u|p−1u in the radial Sobolev space Hr1(RN), where b &amp;gt; 0 and p &amp;gt; 1. We show the global existence and energy scattering in the intercritical regime, i.e., p&amp;gt;N+4+2bN and p&amp;lt;N+2+2bN−2 if N ≥ 3. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo–Nirenberg-type inequalities. Our proofs are based on improved Gagliardo–Nirenberg inequalities, the Morawetz–Sobolev approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)], radial Sobolev embeddings, and localized virial estimates.

Analysis of mass controlled reaction–diffusion systems with nonlinearities having critical growth rates

We analyze semilinear reaction–diffusion systems that are mass controlled, and have nonlinearities that satisfy critical growth rates. The systems under consideration are only assumed to satisfy natural assumptions, namely the preservation of non-negativity and a control of the total mass. It is proved in dimension one that if nonlinearities have (slightly super-) cubic growth rates then the system has a unique global classical solutions. Moreover, in the case of mass dissipation, the solution is bounded uniformly in time in sup-norm. One key idea in the proof is the Hölder continuity of gradient of solutions to parabolic equation with possibly discontinuous diffusion coefficients and low regular forcing terms. When the system possesses additionally an entropy inequality, the global existence and boundedness of a unique classical solution is shown for nonlinearities satisfying a cubic intermediate sum condition, which is a significant generalization of cubic growth rates. The main idea in this case is to combine a modified Gagliardo-Nirenberg inequality and the newly developed L^p-energy method in Fitzgibbon et al. (SIAM J Math Anal 53(6):6771–6803, 2021) and Morgan and Tang (Commun Contemp Math, 2022). This idea also allows us to deal with the case of discontinuous diffusion coefficients in higher dimensions, which has only recently been touched in the context of mass controlled reaction–diffusion systems.

Weighted Gagliardo-Nirenberg interpolation inequalities

In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives, generalizing the celebrated result with classical derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupt-type conditions. We also obtain improved results for some particular families of weights, including power-law weights |x|α. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights 〈x〉α=(1+|x|2)α2.

On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow

Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

A note on inhomogeneous fractional Schrödinger equations

We study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data iu˙−(−Δ)su±|x|ρ|u|p−1u±|u|q−1u=0,0<s<1,ρ≠0,p,q>1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ i\\dot{u}-(-\\Delta )^{s} u \\pm \\vert x \\vert ^{\\rho} \\vert u \\vert ^{p-1}u\\pm \\vert u \\vert ^{q-1}u=0, \\quad 0< s< 1, \\rho \ eq 0, p,q>1. $$\\end{document} Our aim is to treat the competition between the homogeneous term |u|^{q-1}u and the inhomogeneous one |x|^{rho}|u|^{p-1}u. We simultaneously treat two different regimes, rho >0 and rho <-2s. We deal with three technical challenges at the same time: the absence of a scaling invariance, the presence of the singular decaying term |cdot |^{rho}, and the nonlocality of the fractional differential operator (-Delta )^{s}. We give some sufficient conditions on the datum and the parameters N, s, ρ, p, q to have the global versus nonglobal existence of energy solutions. We use the associated ground states and some sharp Gagliardo–Nirenberg inequalities. Moreover, we investigate the L^{2} concentration of the mass-critical blowing-up solutions. Finally, in the attractive regime, we prove the scattering of energy global solutions. Since there is a loss of regularity in Strichartz estimates for the fractional Schrödinger problem with nonradial data, in this work, we assume that u_{|t=0} is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603, 2016).

Concentration and local uniqueness of minimizers for mass critical degenerate Kirchhoff energy functional

In this paper, we consider the L2-norm prescribed minimizer of the mass critical Kirchhoff type energy functional with a weight function a(x),E(u)=∫RNa(x)|∇u|2dx+b2(∫RN|∇u|2dx)2−NN+4∫RN|u|2N+8Ndx,N=1,2,3. Making use of the Gagliardo–Nirenberg inequality, we firstly give the classification of existence and non-existence of minimizers. Then the mass concentration of minimizers as c↗c⁎:=(b‖Q‖28/N2)N8−2N is investigated, where Q>0 is the unique radially symmetric positive solution of 2ΔQ−(4N−1)Q+Q8N+1=0 in RN. It is surprise that the concentrating point of a minimizer is possibly determined by the weight function a(x). Finally, we analyze the local uniqueness of minimizers induced by concentration.

On the maximizing problem associated with critical Sobolev inequality under inhomogeneous constraints

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: http://www.rism.it/doc/Ishiwata.pdf] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities).