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 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
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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
FaberKrahn inequality
I record a proof of the FaberKrahn 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
On the existence of a metric compatible with a given connection
Question: Suppose we are given a torsionfree (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its LeviCivita connection is ? If so, is it unique if we … Continue reading
Posted in Differential equations, Geometry
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A brief introduction to Gammaconvergence
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
Posted in Differential equations
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HopfCole 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
Posted in Differential equations
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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
Posted in Calculus, Differential equations, Geometry
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A problem in analysis
[Updated: 1752011 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
Posted in Analysis, Differential equations
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