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Elliptic equations strongly degenerate at a point. (English) Zbl 1233.35102

Summary: We propose some problems for a class of degenerate elliptic equations, either linear or nonlinear. We study some special cases of these problems and reveal some phenomena which may not have been noticed previously.
Our problems originate from the self-similar solutions of the heat flow of harmonic maps. We prove that the self-similar solutions or the so-called quasi-harmonic spheres are discontinuous at infinity for the equivariant case. In other words, the equivariant quasi-harmonic spheres are not continuous images of topological spheres.

MSC:

35J70 Degenerate elliptic equations
35C05 Solutions to PDEs in closed form
47J30 Variational methods involving nonlinear operators
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58E20 Harmonic maps, etc.
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References:

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