【摘 要】
：
We study bounded solutions of Allen-Cahn equation: -△u = u-u3 in Rn, corresponding to energy functional J(u) =∫|▽u|2+1/2(u2-1)2. A result of Savin states
【机 构】
：
NorthChinaElectricPowerUniversity,China
【出 处】
：
2016年非线性偏微分方程和变分方法及其应用研讨会(Workshop on Nonlinear PDEs and Cal
论文部分内容阅读
We study bounded solutions of Allen-Cahn equation: -△u = u-u3 in Rn, corresponding to energy functional J(u) =∫|▽u|2+1/2(u2-1)2. A result of Savin states that if u is a global minimizer of J, then it is one dimensional, when n < 8. The construction of Del Pino-Kowalczyk-Wei on the other hand gives us examples of nontrivial global minimizers in the case n > 8. However, analogous results in minimal surface theory indicate that nontrivial global minimizers should also exist in dimension 8. We prove this is indeed true, using standard minimization arguments and the family of solutions constructed by Pacard-Wei. We also discuss related questions in a free boundary problem. This is joint work with Kelei Wang and Juncheng Wei.
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