Journal ArticleDOI
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
TLDR
In this paper, the authors show that the sharp Sobolev inequality on Rn can be computed using conformal invariance and geometric symmetrization, and they show that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2).Abstract:
where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see [8]). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is "What is the situation in dimension 2?" Two important arguments ([25], [26], [27]) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in partread more
Citations
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Classification of solutions for an integral equation
TL;DR: In this paper, it was shown that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn.
Journal Article
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
TL;DR: The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it in its simplest form it is a conservative drift-diffusion equation coupled to an elliptic equation for the chemo-attractant concentration as mentioned in this paper.
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Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆
Manuel del Pino,Jean Dolbeault +1 more
TL;DR: In this article, a special class of Gagliardo-nirenberg type inequalities is proposed to interpolate between the classical Sobolev inequality and the Gross logarithmic Soboleve inequality.
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Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary
TL;DR: In this article, the Riemann mapping theorem was generalized to higher dimensions and it was shown that a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways: it has zero scalar curvature and its boundary has constant mean curvature.
References
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Book
Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
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Quantum Geometry of Bosonic Strings
TL;DR: In this article, a formalism for computing sums over random surfaces which arise in all problems containing gauge invariance (like QCD, three-dimensional Ising model etc.) is developed.
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Toeplitz Forms and Their Applications.
TL;DR: In this article, Toeplitz forms are used for the trigonometric moment problem and other problems in probability theory, analysis, and statistics, including analytic functions and integral equations.
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Best constant in Sobolev inequality
TL;DR: The best constant for the simplest Sobolev inequality was proved in this paper by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.