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Register a new userSup-Inf inequality on manifold of dimension 3
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Authors
Samy Skander Bahoura
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We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).
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We give a sup+inf inequality on $S_4$ for Paneitz operator.
This paper is in relation with a Note of Comptes Rendus de lAcademie des Sciences 2005. We have an idea about a lower bounds of sup+inf (2 dimensions) and sup*inf (dimensions >2).
In this paper, we study the extremal problem for the Strichartz inequality for the Schr{o}dinger equation on $mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.
Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(Sigma,g)$. Precisely, if $lambda_1(Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $alpha^ast<lambda_1(Sigma)$ such that for all $alphain (alpha^ast,lambda_1(Sigma))$, the supremum $$sup_{uin W^{1,2}(Sigma,g),,int_Sigma udv_g=0,,| abla_gu|_2leq 1}int_Sigma exp(4pi u^2(1+alpha|u|_2^2))dv_g$$ can not be attained by any $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $| abla_gu|_2leq 1$, where $W^{1,2}(Sigma,g)$ denotes the usual Sobolev space and $|cdot|_2=(int_Sigma|cdot|^2dv_g)^{1/2}$ denotes the $L^2(Sigma,g)$-norm. This complements our earlier result in [39].
In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrodinger flow in the whole space $R^n$ provided there is a smooth solution in the energy class.
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