论文标题
密度周长的非本地等级问题
A nonlocal isoperimetric problem with density perimeter
论文作者
论文摘要
我们认为，在体积约束下，由密度周长和RIESZ类型与指数$α$的非定位相互作用给出的能量功能最小化，其中非局部相互作用的强度由参数$γ$控制。我们表明，对于广泛的密度，能量可容纳任何值$γ$的最小化器。此外，这些最小化器是有限的。对于$ | x |^p $的单一密度，我们证明，当$γ$足够小时，唯一的最小化器将由固定体积的球给出。与恒定密度的情况相反，此处$γ\ to 0 $限制在适当的重新缩放下与小质量$ m = |ω| \至0 $限制时，当$ p <d-d-α+1 $时，但与powers $ p> p> d-α+1 $ 1 $相对应。
We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $α$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $γ$. We show that for a wide class of density functions the energy admits a minimizer for any value of $γ$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $γ$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $γ\to 0$ limit corresponds, under a suitable rescaling, to a small mass $m=|Ω|\to 0$ limit when $p<d-α+1$, but to a large mass $m\to\infty$ for powers $p>d-α+1$.
