Existence Of Extremal Functions Research Articles

Singular Trudinger–Moser inequality with Lp norm and the existence of extremals

In this paper, we establish the singular Trudinger–Moser inequality with norm in a bounded domain and the existence of extremal functions by blow-up analysis. As our first main result, we prove that for , , and , the singular Trudinger–Moser inequality holds, where is the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality with norm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.

The existence of extremal functions for discrete Sobolev inequalities on lattice graphs

In this paper, we study the existence of extremal functions (pairs) of the following discrete Sobolev inequality (0.1) and Hardy-Littlewood-Sobolev inequality (0.2) in the lattice ZN:(0.1)‖u‖ℓq≤Cp,q‖u‖D1,p,∀u∈D1,p(ZN), where N≥3,1≤p<N,q>p⁎=NpN−p,Cp,q is a constant depending on N,p and q;(0.2)∑i,j∈ZNi≠jf(i)g(j)|i−j|λ≤Cr,s,λ‖f‖ℓr‖g‖ℓs,∀f∈ℓr(ZN),g∈ℓs(ZN), where r,s>1, 0<λ<N, 1r+1s+λN>2, Cr,s,λ is a constant depending on N,r,s and λ.We introduce the discrete Concentration-Compactness principle, and prove the existence of extremal functions (pairs) for the best constants in the supercritical cases q>p⁎ and 1r+1s+λN>2, respectively.

Sharp exponential inequalities for the Ornstein-Uhlenbeck operator

The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.

Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality in the spirit of Tintarev (J. Funct. Anal. 266, 55–66, 2014) and prove the existence of extremal functions for this improved singular Moser–Trudinger inequality. Our proof is based on the blow-up analysis method. Due to appearance of the singularity of weights, our proof is more complicated and difficult in dealing with analyzing the asymptotic behavior of the sequence of maximizers in the subcritical case near the blow-up point when comparing with the previous works (see, e.g., Nguyen (Ann. Global Anal. Geom. 54(2), 237–256, 2018), Yang (J. Funct. Anal. 239(1), 100–126, 2006)). To overcome these difficulties, we shall prove a classification result for a singular quasi-linear Liouville equation which maybe is of independent interest. Finally, we derive another improvement of the singular Moser–Trudinger inequality in the sprit of Adimurthi and Druet (Comm. Partial Differential Equations 29, 295–322, 2004). Our results extend many well-known Moser–Trudinger type inequalities to more general setting (e.g., singular weights, higher dimension, any domain, etc).

Extremal functions for Trudinger\u2013Moser inequalities with nonnegative weights

Using blow-up analysis, the author proves the existence of extremal functions for Trudinger–Moser inequalities with nonnegative weights on bounded Euclidean domains or compact Riemannian surfaces. This extends recent results of Yang (J. Differ. Equ. 258:3161–3193, 2015) and Yang–Zhu (Proc. Am. Math. Soc. 145:3953–3959, 2017).

Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions

The main purpose of this paper is to establish sharp weighted Trudinger–Moser inequalities (Theorems 1.1, 1.2 and 1.3) and Caffarelli–Kohn–Nirenberg inequalities in the borderline case p=N (Theorems 1.5, 1.6 and 1.7) with best constants. Existence of extremal functions is also investigated for both the weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities. Radial symmetry of extremal functions for the weighted Trudinger–Moser inequalities are established (Theorem 1.4). Moreover, the sharp constants and the forms of the optimizers for the Caffarelli–Kohn–Nirenberg inequalities in some particular families of parameters in the borderline case p=N will be computed explicitly. Symmetrization arguments do not work in dealing with these weighted inequalities because of the presence of weights and the failure of the Polyá – Szegö inequality with weights. We will thus use a quasi-conformal mapping type transform and the corresponding symmetrization lemma to overcome this difficulty and carry out proofs of these results. As an application of the Caffarelli–Kohn–Nirenberg inequality, we also establish a weighted Moser–Onofri type inequality on the entire Euclidean space R2 (see Theorem 1.8).

Equivalence of optimal [formula omitted]-inequalities on Riemannian manifolds

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥2. This paper concerns the validity of the optimal Riemannian L1-Entropy inequalityEntdvg(u)≤nlog⁡(Aopt‖Du‖BV(M)+Bopt) for all u∈BV(M) with ‖u‖L1(M)=1 and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent to an optimal L1-Sobolev inequality obtained by Druet [3].

Sharp Nash inequalities on the unit sphere: The influence of symmetries

In this paper both we establish the best constants for the Nash inequalities on the standard unit sphere S n of R n + 1 , n ≥ 3 and we give answers on the existence of extremal functions on the corresponding problems. Also we study the problem of the best constants in the case where the data are invariant under the action of the group G = O ( k ) × O ( m ) , k + m = n + 1 and we find the best constants.

A necessary and sufficient condition for existence of extremal functions of a linear functional on H 1

We consider the following well-known old problem: (1) find a necessary and sufficient condition for existence of the functions on the unit sphere of the Hardy space (p = 1) at which the norm of a linear functional is attained; (2) obtain parametric description of the set of these functions; (3) find conditions for uniqueness of the extremal functions. We prove that the answers to these questions follow from existence and uniqueness of a solution to a homogeneous linear integral equation whose kernel is explicit in terms of the function determining the analytical representation of the indicated linear functional. Its extremal functions can be obtained from solutions to this integral equation.

A sharp form of trace Moser–Trudinger inequality on compact Riemannian surface with boundary

In this paper, a sharp form of trace Moser–Trudinger inequality is established on the boundary of compact Riemannian surface, and the existence of extremal function is proved via the method of blowing up analysis.

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.

Quadratic Extremal Problems on the Dirichlet Space

A class of quadratic extremal problems over the unit ball of the Dirichlet space is studied. Existence of extremal functions is established. The boundary regularity of solutions is determined in terms of the regularity of the underlying functional. It is further shown that the extremal functions are entire functions in a range of cases.

Existence of extremal functions in inequalities for derivatives

Existence of extremal functions in inequalities for derivatives