In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain does not allow balls of arbitrarily large radius (finite ball condition). We prove that such a result can not be true in the `nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all eigenvalues of fractional Dirichlet problems on long cylindrical domains is addressed.
The central aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s \in (0,1)$. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains $(ω\times \mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [\text…
The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincaré domains). We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $r\in[1,3]$) perturbed by an additive noise on Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these…
In this work, we carry out the asymptotic analysis of the two dimensional convective Brinkman-Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and Poincaré domains (using asymptotic compactness property). In Poincaré domains, for $r=1,2$ and $3$, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors for the 2D CBF equations. We con…
Let $Ω$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $μ_1(Ω)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $Ω$; we prove that $μ_1(Ω) \ge 1$. The result is sharp since equality sign is achieved when $Ω$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(Ω,dγ_N)$, where $γ_N$ is the $N$-dimensional Gaussian measure.
Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains
Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Technology (NOVA SST), Portugal（NOVA科学与技术学院数学与应用中心（NOVA Math））；Department of Mathematics, NOVA School of Science and Technology (NOVA SST), Portugal（NOVA科学与技术学院数学系）
We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.
The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces which restrict the choice of the initial data. Using the "tail estimation method" introduced by B. Wang for reaction diffusion equations, we establish in this paper the existence of a global attractor for two wave equations in the traditional Hilbert spaces $\displaystyle H^1(Ω)\times L^2(Ω)$ where $Ω$ is an unbounded domain of $\R^N$. The first equation, with…
Random dynamics of solutions for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded Poincaré domains
Tata Institute of Fundamental Research - Centre For Applicable Mathematics（塔塔基础研究中心应用数学中心）；Center for Mathematics and Applications (NovaMath), Universidade Nova de Lisboa（新里斯本大学数学与应用中心（NovaMath））；Department of Mathematics, Indian Institute of Technology Roorkee（印度理工学院鲁尔基分校数学系）
We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.
The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise,…
This paper investigates the link between the Maximum Principle and the sign of the (generalized) principal eigenvalue for elliptic operators in unbounded domains. Our approach covers the cases of Dirichlet, Neumann, and (indefinite) Robin boundary conditions and treat them in a unified way. For a certain class of elliptic operators (including the class of selfadjoint operators), we establish that the positivity of the principal eigenvalue is a necessary and sufficient condition for the validity of the Maximum Principle. If the principal eigenvalue is zero, no general answer holds; instead, und…
In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…
Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains
Hanoi Pedagogical University（河内师范大學）；Electric Power University（電力大學）；Tata Institute of Fundamental Research - Centre For Applicable Mathematics (TIFR-CAM)（塔塔基礎研究所應用數學中心）；NOVA School of Science and Technology (NOVA FCT)（新里斯本科技學院）；Indian Institute of Technology Roorkee-IIT Roorkee（魯爾凱印度理工學院）
On the Bakry-Emery criterion for linear diffusions and weighted porous media equations
给出了广义Poincaré不等式的一个非局部充分条件，推广了著名的Bakry-Emery条件. W. Beckner在高斯情形下的结果，并沿着Ornstein-Uhlenbeck流，给出了在$L^2$范数和通常熵之间插值的某些广义熵的指数衰减。我们的标准改进的结果，例如，可以推导出从Bakry-Emery准则和Holley-Stroock型扰动结果。在第二步中，我们将相同的策略应用于多孔介质类型的非线性方程。这为发展问题的解提供了新的插值不等式和衰减估计。该标准再次是一个非局部条件的基础上的积极性的最低本征值的薛定谔算子。在这两种情况下，我们将Fisher信息与其时间导数联系起来。由于所得到的标准是非本地的，它是更好地适应的潜力，例如，在无穷大的非二次增长，或无界扰动的潜力。
In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially for unbounded Laplacians. By proving Poincaré inequality, we obtain a lower bound on Cheeger constant in terms of positive curvature.
In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behaviour of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-i…
