اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTrudinger-Moser inequality on the whole plane with the exact growth condition
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Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modifie
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We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, b
etageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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 approac
h to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.
In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $mathbb{B}$ be the unit ball in $mathbb{R}^N$ $(Ngeq 2)$, $p>1$, $g=|x|^{frac{2p}{N}beta}(dx_1^2+cdots+dx_N^2)$ be a conical metric on $mathbb{B}$
, and $lambda_p(mathbb{B})=infleft{int_mathbb{B}| abla u|^Ndx: uin W_0^{1,N}(mathbb{B}),,int_mathbb{B}|u|^pdx=1right}$. We prove that for any $betageq 0$ and $alpha<(1+frac{p}{N}beta)^{N-1+frac{N}{p}}lambda_p(mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $uin W_0^{1,N}(mathbb{B})$ with $int_mathbb{B}| abla u|^Ndx-alpha(int_mathbb{B}|u|^p|x|^{pbeta}dx)^{N/p}leq 1$, there holds $$int_mathbb{B}e^{alpha_N(1+frac{p}{N}beta)|u|^{frac{N}{N-1}}}|x|^{pbeta}dxleq C,$$ where $|x|^{pbeta}dx=dv_g$, $alpha_N=Nomega_{N-1}^{1/(N-1)}$, $omega_{N-1}$ is the area of the unit sphere in $mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<beta<0$ and $alpha=0$ was considered by Adimurthi-Sandeep cite{A-S}, while the case $p=N=2$, $betageq 0$ and $alpha=0$ was studied by de Figueiredo-do O-dos Santos cite{F-do-dos}.
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}(Om
ega), ] 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 in
equalities 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.
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