The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Maochun Zhu - One of the best experts on this subject based on the ideXlab platform.
-
existence and nonexistence of Extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2
Advances in Mathematics, 2020Co-Authors: Lu Chen, Maochun ZhuAbstract:Abstract Though much progress has been made with respect to the existence of Extremals of the critical first order Trudinger-Moser inequalities in W 1 , n ( R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R 4 . The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31] ), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the form S ( α ) = sup ‖ u ‖ H 2 = 1 ∫ R 4 ( exp ( 32 π 2 | u | 2 ) − 1 − α | u | 2 ) d x , where α ∈ ( − ∞ , 32 π 2 ) . We establish the existence of the threshold α ⁎ , where α ⁎ ≥ ( 32 π 2 ) 2 B 2 2 and B 2 ≥ 1 24 π 2 , such that S ( α ) is attained if 32 π 2 − α α ⁎ , and is not attained if 32 π 2 − α > α ⁎ . This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
-
existence and nonexistence of Extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2
arXiv: Analysis of PDEs, 2018Co-Authors: Lu Chen, Maochun ZhuAbstract:Though much work has been done with respect to the existence of Extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}(\mathbb{R}^n)$ and higher order Adams inequalities on finite domain $\Omega\subset \mathbb{R}^n$, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space $\mathbb{R}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego\ type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see \cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in $\mathbb{R}^4$ of the form $$ S(\alpha)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32\pi^2|u|^2)-1-\alpha|u|^2\big)dx,$$ where $\alpha \in (-\infty, 32\pi^2)$. We establish the existence of the threshold $\alpha^{\ast}$, where $\alpha^{\ast}\geq \frac{(32\pi^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24\pi^2}$, such that $S\left( \alpha\right) $ is attained if $32\pi^{2}-\alpha \alpha^{\ast}$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on $\mathbb{R}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
Lu Chen - One of the best experts on this subject based on the ideXlab platform.
-
existence and nonexistence of Extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2
Advances in Mathematics, 2020Co-Authors: Lu Chen, Maochun ZhuAbstract:Abstract Though much progress has been made with respect to the existence of Extremals of the critical first order Trudinger-Moser inequalities in W 1 , n ( R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R 4 . The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31] ), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the form S ( α ) = sup ‖ u ‖ H 2 = 1 ∫ R 4 ( exp ( 32 π 2 | u | 2 ) − 1 − α | u | 2 ) d x , where α ∈ ( − ∞ , 32 π 2 ) . We establish the existence of the threshold α ⁎ , where α ⁎ ≥ ( 32 π 2 ) 2 B 2 2 and B 2 ≥ 1 24 π 2 , such that S ( α ) is attained if 32 π 2 − α α ⁎ , and is not attained if 32 π 2 − α > α ⁎ . This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
-
sharp weighted trudinger moser adams inequalities on the whole space and the existence of their Extremals
Calculus of Variations and Partial Differential Equations, 2019Co-Authors: Lu Chen, Caifeng ZhangAbstract:Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’ inequalities. In this paper, we mainly study the existence of Extremals for sharp weighted Trudinger–Moser–Adams type inequalities with the Dirichlet and Sobolev norms (also known as the critical and subcritical Trudinger–Moser–Adams inequalities), see Theorems 1.1, 1.2, 1.3, 1.5, 1.7, 1.9 and 1.11. First, we employ the method based on level-sets of functions under consideration and Fourier transform to establish stronger weighted Trudinger–Moser–Adams type inequalities with the Dirichlet norm in $$W^{2,\frac{n}{2}}(\mathbb {R}^n)$$ and $$W^{m,2}(\mathbb {R}^{2m})$$ respectively. While the first order sharp weighted Trudinger–Moser inequality and its existence of extremal functions was established by Dong and the second author using a quasi-conformal type transform (Dong and Lu in Calc Var Partial Differ Equ 55:55–88, 2016), such a transform does not work for the Adams inequality involving higher order derivatives. Since the absence of the Polya–Szego inequality and the failure of change of variable method for higher order derivatives for weighted inequalities, we will need several compact embedding results (Lemmas 2.1, 3.1 and 5.2). Through the compact embedding and scaling invariance of the subcritical Adams inequality, we investigate the attainability of best constants. Second, we employ the method developed by Lam et al. (Adv Math 352:1253–1298, 2019) which uses the relationship between the supremums of the critical and subcritical inequalities (see also Lam in Proc Amer Math Soc 145:4885–4892, 2017) to establish the existence of Extremals for weighted Adams’ inequalities with the Sobolev norm. Third, using the Fourier rearrangement inequality established by Lenzmann and Sok (A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, arXiv:1805.06294v1 ), we can reduce our problem to the radial case and then establish the existence of the extremal functions for the non-weighted Adams inequalities. As an application, we derive new results on high-order critical Caffarelli–Kohn–Nirenberg interpolation inequalities whose parameters extend those proved by Lin (Commun Partial Differ Equ 11:1515–1538, 1986) (see Theorems 1.13 and 1.14). Furthermore, we also establish the relationship between the best constants of the weighted Trudinger–Moser–Adams type inequalities and the Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorems 1.13 and 1.14).
-
existence and nonexistence of Extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2
arXiv: Analysis of PDEs, 2018Co-Authors: Lu Chen, Maochun ZhuAbstract:Though much work has been done with respect to the existence of Extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}(\mathbb{R}^n)$ and higher order Adams inequalities on finite domain $\Omega\subset \mathbb{R}^n$, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space $\mathbb{R}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego\ type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see \cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in $\mathbb{R}^4$ of the form $$ S(\alpha)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32\pi^2|u|^2)-1-\alpha|u|^2\big)dx,$$ where $\alpha \in (-\infty, 32\pi^2)$. We establish the existence of the threshold $\alpha^{\ast}$, where $\alpha^{\ast}\geq \frac{(32\pi^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24\pi^2}$, such that $S\left( \alpha\right) $ is attained if $32\pi^{2}-\alpha \alpha^{\ast}$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on $\mathbb{R}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
Lisa Randall - One of the best experts on this subject based on the ideXlab platform.
-
extremal limits and black hole entropy
Journal of High Energy Physics, 2009Co-Authors: Sean M Carroll, Matthew C Johnson, Lisa RandallAbstract:Taking the extremal limit of a non-extremal Reissner-Nordstrom black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate — while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of AdS_2 × S^2. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordstrom solution splits into two spacetimes at extremality: an extremal black hole and a disconnected AdS space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.
-
extremal limits and black hole entropy
arXiv: High Energy Physics - Theory, 2009Co-Authors: Sean M Carroll, Matthew C Johnson, Lisa RandallAbstract:Taking the extremal limit of a non-extremal Reissner-Nordstr\"om black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate -- while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of $AdS_2\times S^2$. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordstr\"om solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.
Y Zhang - One of the best experts on this subject based on the ideXlab platform.
-
geometric optimal control of the contrast imaging problem in nuclear magnetic resonance
IEEE Transactions on Automatic Control, 2012Co-Authors: Bernard Bonnard, Olivier Cots, Steffen J Glaser, M Lapert, D Sugny, Y ZhangAbstract:The objective of this article is to introduce the tools to analyze the contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories can be selected among extremal solutions of the Pontryagin Maximum Principle applied to this Mayer type optimal problem. Such trajectories are associated to the question of extremizing the transfer time. Hence the optimal problem is reduced to the analysis of the Hamiltonian dynamics related to singular Extremals and their optimality status. This is illustrated by using the examples of cerebrospinal fluid/water and grey/white matter of cerebrum.
Bernard Bonnard - One of the best experts on this subject based on the ideXlab platform.
-
geometric optimal control of the contrast imaging problem in nuclear magnetic resonance
IEEE Transactions on Automatic Control, 2012Co-Authors: Bernard Bonnard, Olivier Cots, Steffen J Glaser, M Lapert, D Sugny, Y ZhangAbstract:The objective of this article is to introduce the tools to analyze the contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories can be selected among extremal solutions of the Pontryagin Maximum Principle applied to this Mayer type optimal problem. Such trajectories are associated to the question of extremizing the transfer time. Hence the optimal problem is reduced to the analysis of the Hamiltonian dynamics related to singular Extremals and their optimality status. This is illustrated by using the examples of cerebrospinal fluid/water and grey/white matter of cerebrum.
-
nuclear magnetic resonance the contrast imaging problem
Conference on Decision and Control, 2011Co-Authors: Bernard Bonnard, Steffen J Glaser, Monique Chyba, John Marriott, D SugnyAbstract:Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal problem and the Pontryagin Maximum Principle is used to select the optimal trajectories among the extremal solutions. Correlation between the contrast problem and the optimal transfer time problem is demonstrated. Further, we develop some analysis of the singular Extremals and apply the results to examples of cerebrospinal fluid/water and grey/white matter of the cerebrum.
-
riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust
Annales De L Institut Henri Poincare-analyse Non Lineaire, 2007Co-Authors: Bernard Bonnard, Jeanbaptiste CaillauAbstract:Abstract This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion and is based on preliminary results of Epenoy et al. (1997) where the optimal trajectories of the energy minimization problem are approximated using averaging techniques. The averaged Hamiltonian system is explicitly computed. It is related to a Riemannian problem whose distance is an approximation of the value function. The extremal curves are analyzed, proving that the system remains integrable in the coplanar case. It is also checked that the metric associated with coplanar transfers towards a circular orbit is flat. Smoothness of small Riemannian spheres ensures global optimality of the computed Extremals.
