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arXiv · 1404.3904

On the growth of the energy of entire solutions to the vector Allen-Cahn equation

Christos Sourdis
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Abstract

We prove that the energy over balls of entire, nonconstant, bounded solutions to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for any $k>0$, as the volume $R^n$ of the ball tends to infinity. This improves the growth rate of order $R^{n-2}$ that follows from the general weak monotonicity formula. Moreover, our estimate can be considered as an approximation to the corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.

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Christos Sourdis. 2014-04-25. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. https://arxiv.org/abs/1404.3904

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