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

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  • 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 radius $R$ of the $n$-dimensional ball tends to infinity. This improves the growth rate of order $R^{n-2}$ if $n\geq 3$ and $\ln R$ if $n=2$ that follows from the general weak monotonicity formula. Moreover, our estimate may be considered as an approximation to the corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.
    Mathematics Subject Classification: Primary: 35J20, 35J91; Secondary: 35J47.


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