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Register a new userAn inequality of type sup+inf on $S_4$ for the Paneitz operator
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Authors
Samy Skander Bahoura
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We give a sup+inf inequality on $S_4$ for Paneitz operator.
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We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).
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 article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Delta^2$ on a bounded smooth domain $Om$ in the Euclidean $n$-space ${bf R}^n$ ($nge2$) and then prove that the corresponding first non-zero eigenvalue $Upsilon_1(Om)$ admits the isoperimetric inequality of Szego-Weinberger type: $Upsilon_1(Om)le Upsilon_1(B_{Om})$, where $B_{Om}$ is a ball in ${bf R}^n$ with the same volume of $Om$. The isoperimetric inequality of Szego-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Delta^{2m}$ ($mge1$) on $Om$ is also exploited.
In this paper we consider a fourth order equation involving the critical Sobolev exponent on a bounded and smooth domain in $R^6$. Using theory of critical points at infinity, we give some topological conditions on a given function defined on a domain to ensure some existence results.
We derive the first and second variation formula for the Greens function poles value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the second eigenvalue and Sobolev inequality of Paneitz operator.
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