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Register a new userSharp Moser-Trudinger inequalities on Riemannian manifolds with Negative curvature
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Let $M$ be a complete, simply connected Riemannian manifold with negative curvature. We obtain some Moser-Trudinger inequalities with sharp constants on $M$.
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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 $R^2$.
Let $(Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(Sigma,g)$ be the usual Sobolev space, $textbf{G}$ be a finite isometric group acting on $(Sigma,g)$, and $mathscr{H}_textbf{G}$ be a function space including all functions $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $u(sigma(x))=u(x)$ for all $sigmain textbf{G}$ and all $xinSigma$. Denote the number of distinct points of the set ${sigma(x): sigmain textbf{G}}$ by $I(x)$ and $ell=inf_{xin Sigma}I(x)$. Let $lambda_1^textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $mathscr{H}_textbf{G}$. Using blow-up analysis, we prove that if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then there holds $$sup_{uinmathscr{H}_textbf{G},,int_Sigma| abla_gu|^2dv_g-alpha int_Sigma u^2dv_gleq 1}int_Sigma e^{beta u^2}dv_g<infty;$$ if $alpha<lambda_1^textbf{G}$ and $beta>4piell$, or $alphageq lambda_1^textbf{G}$ and $beta>0$, then the above supremum is infinity; if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser cite{Moser}, L. Fontana cite{Fontana} and W. Chen cite{Chen-90}.
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 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.
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].
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