Category Archives: Differential equations

Weighted isoperimetric inequalities in warped product manifolds

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading

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Faber-Krahn inequality

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading

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On the existence of a metric compatible with a given connection

Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its Levi-Civita connection is ? If so, is it unique if we … Continue reading

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A brief introduction to Gamma-convergence

1. A motivating example Consider the -Laplace equation in , where is a nonempty bounded open subset of , and on . The energy functional associated with the PDE is . For , because of convexity, has a unique minimizer … Continue reading

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Hopf-Cole transformation and the Burgers’ equation

The Burgers equation is the following: Here can be regarded as the (scalar) velocity of a fluid and can be regarded as the viscosity. This equation can be regarded as the simplified version of the Euler’s equation without the pressure … Continue reading

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Reilly type formula and its applications II

This is a sequel to my previous post Reilly type formula and its applications. (Oops… it’s been a long time!) 1. Introduction Integral formulas have always been an important tool for studying various analytical and geometric problems on Riemannian manifolds. … Continue reading

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A problem in analysis

[Updated: 17-5-2011 for a natural corollary. ] Last week, when I was giving a seminar I encountered a problem in analysis (which I was unable to solve at that time). After discussing with John Ma, we have come up with … Continue reading

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