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Register a new userAubin type almost sharp Moser-Trudinger inequality revisited
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Fengbo Hang
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We give a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces. In particular we can lower the smoothness requirement of the metric and apply the same approach to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.
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Wang and Ye conjectured in [22]: Let $Omega$ be a regular, bounded and convex domain in $mathbb{R}^{2}$. There exists a finite constant $C({Omega})>0$ such that [ int_{Omega}e^{frac{4pi u^{2}}{H_{d}(u)}}dxdyle C(Omega),;;forall uin C^{infty}_{0}(Omega), ] where $H_{d}=int_{Omega}| abla u|^{2}dxdy-frac{1}{4}int_{Omega}frac{u^{2}}{d(z,partialOmega)^{2}}dxdy$ and $d(z,partialOmega)=minlimits_{z_{1}inpartialOmega}|z-z_{1}|$.} The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $mathbb{R}^{2}$ via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space $mathbb{B}={z=x+iy:|z|=sqrt{x^{2}+y^{2}}<1}$: [ sup_{|u|_{mathcal{H}}leq 1} int_{mathbb{B}}(e^{4pi u^{2}}-1-4pi u^{2})dV=sup_{|u|_{mathcal{H}}leq 1}int_{mathbb{B}}frac{(e^{4pi u^{2}}-1-4pi u^{2})}{(1-|z|^{2})^{2}}dxdy< infty, ] by using the method employed earlier by Lam and the first author [9, 10], where $mathcal{H}$ denotes the closure of $C^{infty}_{0}(mathbb{B})$ with respect to the norm $$|u|_{mathcal{H}}=int_{mathbb{B}}| abla u|^{2}dxdy-int_{mathbb{B}}frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].
A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin type inequalities on $mathbb{S}^{1}$ coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method.
Let $W^{1,n} ( mathbb{R}^{n} $ be the standard Sobolev space and $leftVert cdotrightVert _{n}$ be the $L^{n}$ norm on $mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm [ underset{leftVert urightVert _{W^{1,n}left(mathbb{R} ^{n}right) }=1}{sup}int_{ mathbb{R}^{n}}Phileft( alpha_{n}leftvert urightvert ^{frac{n}{n-1}}left( 1+alphaleftVert urightVert _{n}^{n}right) ^{frac{1}{n-1}}right) dx<+infty ]in the entire space $mathbb{R}^n$ for any $0leqalpha<1$, where $Phileft( tright) =e^{t}-underset{j=0}{overset{n-2}{sum}}% frac{t^{j}}{j!}$, $alpha_{n}=nomega_{n-1}^{frac{1}{n-1}}$ and $omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $mathbb{R}^n$. We also show that the above supremum is infinity for all $alphageq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $alpha>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Phi(t)$ is replaced by a strictly smaller $Phi^*(t)=e^{t}-underset{j=0}{overset{n-1}{sum}}% frac{t^{j}}{j!}$. (Note that $Phi(t)=Phi^*(t)+frac{t^{n-1}}{(n-1)!}$).
In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities.
We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Aperys constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.
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