Improved Moser-Trudinger Inequality Involving Lp Norm in n Dimensions

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Jiuyi Zhu
Keyword(s):  
Blow Up ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Lp Norm ◽  
Blow Up Analysis ◽  
Trudinger Inequality

AbstractThe paper is concerned about an improvement of Moser-Trudinger inequality involving Lbe the first eigenvalue associated with n-Laplacian. We obtain the following strengthened Moser-Trudinger inequality with blow-up analysisfor 0 ≤ α < λ̅(Ω) and 1 < p ≤ n, and the supremum is infinity for α ≥ λ̅(Ω), where

Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)

2019 ◽  
Vol 21 (04) ◽  
pp. 1850023 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Critical Case ◽  
Blow Up ◽  
Sharp Inequality ◽  
Extremal Functions ◽  
Analysis Method ◽  
Subcritical Case ◽  
Best Constant ◽  
Blow Up Analysis ◽  
Nirenberg Inequality ◽  
Trudinger Inequality

We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

scholarly journals Improved Adams-type inequalities and their extremals in dimension 2m

2020 ◽  
pp. 2050043
Author(s):  
Azahara DelaTorre ◽  
Gabriele Mancini
Keyword(s):  
Sobolev Space ◽  
Open Subset ◽  
Extremal Function ◽  
Blow Up ◽  
New Techniques ◽  
Blow Up Analysis ◽  
Bounded Smooth ◽  
Trudinger Inequality

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

EXTREMAL FUNCTIONS FOR A SHARP MOSER–TRUDINGER INEQUALITY

2006 ◽  
Vol 17 (03) ◽  
pp. 331-338 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Domain ◽  
Extremal Functions ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Trudinger Inequality

Let Ω be a bounded smooth domain in ℝ2, and λ1(Ω) the first eigenvalue of the Laplacian with Dirichlet boundary condition in Ω. Then Adimurthi and Druet show that for any 0 ≤ α < λ1(Ω)[Formula: see text] We prove in this paper that there exist extremal functions for the above inequality. In other words, we show that [Formula: see text] is attained for any 0 ≤ α < λ1(Ω).

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals Lichnerowicz-Obata Estimate, Almost Parallel p-form and Almost Product Manifolds

2021 ◽  
Author(s):  
Masayuki Aino
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Hausdorff Approximation

AbstractWe show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of n-dimensional closed Riemannian manifolds with an almost parallel p-form ($$2\le p \le n/2$$ 2 ≤ p ≤ n / 2 ) in $$L^2$$ L 2 -sense, and give a Gromov-Hausdorff approximation to a product $$S^{n-p}\times X$$ S n - p × X under some pinching conditions when $$2\le p<n/2$$ 2 ≤ p < n / 2 .

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

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