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Nonlinear Analysis

Solutions of the Minimal Surface Equation and of the Monge-Ampere Equation near Infinity

Qing Han, University of Notre Dame

Location:  Zoom
Date & time: Wednesday, 21 April 2021 at 9:30AM - 10:30AM


Abstract: Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation (due to Bers and Schoen) and to quadratic polynomials for the Monge-Ampere equation (due to Caffarelli-Li) for dimension n at least 3, with an extra logarithmic term for n=2. We characterize remainders in the asymptotic expansions by a single function, which is given by a solution of some elliptic equation near the origin via the Kelvin transform. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge-Ampere equation in even dimension, but only C^{n-1,alpha} for the Monge-Ampere equation in odd dimension, for any alpha in (0,1).

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