
Previous Article
ContractionGalerkin method for a semilinear wave equation
 CPAA Home
 This Issue

Next Article
On the dynamics of flows on compact metric spaces
Optimal Hardy inequalities for general elliptic operators with improvements
1.  Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2, Canada 
$ \int_\Omega  \nabla u_A^2 dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2}u^2dx, \qquad u \in H_0^1(\Omega) \qquad\qquad (1)$
where $ E$ is a positive function defined in $ \Omega$, div$(A \nabla E)$ is a nonnegative nonzero finite measure in $ \Omega$ which we denote by $ \mu$ and where $ A(x)$ is a $ n \times n$ symmetric, uniformly positive definite matrix defined in $ \Omega$ with $  \xi _A^2:= A(x) \xi \cdot \xi$ for $ \xi \in \mathbb{R}^n$. We show that (1) is optimal if $ E=0$ on $ \partial \Omega$ or $ E=\infty$ on the support of $ \mu$ and is not attained in either case. When $ E=0$ on $\partial \Omega$ we show
$ \int_\Omega  \nabla u_A^2dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2}u^2dx + \frac{1}{2} \int_\Omega \frac{u^2}{E} d \mu, \qquad u \in H_0^1(\Omega)\qquad (2) $
is optimal and not attained.
Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for $p$≠ 2 which, in the case that $ \mu$ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]).
We examine improved versions of the above inequalities of the form
$\int_\Omega  \nabla u_A^2dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$
Necessary and sufficient conditions on $V$ are obtained (in terms of the solvability of a linear pde)
for (3) to hold. Analogous results involving improvements are obtained for the weighted versions.
In addition we obtain various results concerning the above inequalities valid for functions $ u$ which are nonzero on the boundary of $ \Omega$. We also examine the nonquadradic case ,ie. $p$ ≠2 of the above inequalities.
[1] 
Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405425. doi: 10.3934/cpaa.2020274 
[2] 
Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020469 
[3] 
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems  S, 2021, 14 (1) : 243271. doi: 10.3934/dcdss.2020213 
[4] 
José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369388. doi: 10.3934/cpaa.2020271 
[5] 
Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020170 
[6] 
D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems  S, 2021, 14 (1) : 205217. doi: 10.3934/dcdss.2020346 
[7] 
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : . doi: 10.3934/cpaa.2020272 
[8] 
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixednorm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 54875508. doi: 10.3934/cpaa.2020249 
[9] 
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020462 
[10] 
Anton A. Kutsenko. Isomorphism between oneDimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359368. doi: 10.3934/cpaa.2020270 
[11] 
Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : . doi: 10.3934/era.2020117 
[12] 
Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020048 
[13] 
Adel M. AlMahdi, Mohammad M. AlGharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389404. doi: 10.3934/cpaa.2020273 
[14] 
Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 5997. doi: 10.3934/jcd.2021004 
[15] 
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems  A, 2021, 41 (1) : 277296. doi: 10.3934/dcds.2020138 
[16] 
Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasiperiodic jacobi operators with BrjunoRüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 53055335. doi: 10.3934/cpaa.2020240 
[17] 
A. M. Elaiw, N. H. AlShamrani, A. AbdelAty, H. Dutta. Stability analysis of a general HIV dynamics model with multistages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020441 
[18] 
Yichen Zhang, Meiqiang Feng. A coupled $ p $Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 14191438. doi: 10.3934/era.2020075 
[19] 
Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems  S, 2020 doi: 10.3934/dcdss.2020436 
[20] 
Hoang The Tuan. On the asymptotic behavior of solutions to timefractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems  B, 2020 doi: 10.3934/dcdsb.2020318 
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]