本文证明了问题$$\min\Big\{P(Ω)+\mathcal{G}(Ω):\ Ω\subset D,\ |Ω|=m\Big\},$$最优形状的存在性和正则性，其中$P$表示周长，$|\cdot|$是体积，并且函数$\mathcal{G}$是以下之一：\textless{} ul \textgreater{} \textless{} li \textgreater{}是Dirichlet能量$E\_f$，相对于a（可能改变符号）函数$f\in L^p$ ; \textless{} /li \textgreater{} \textless{} li \textgreater{} $F(λ\_{1},\dots,λ\_{k})$形式的谱泛函，其中$λ\_k$是Dirichlet Laplacian的第$k$个特征值，并且$F:\mathbb{R}^k\to\mathbb{R}$是Lipschitz连续的并且在每个变量中增加。\textless{} /li \textgreater{} \textless{} /ul \textgreater{}域$D$是整个空间$\mathbb{R}^d$或有界域。我们也给出了一般假设的功能$\mathcal{G}$，使结果仍然有效。
英文摘要：
We prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(Ω)+\mathcal{G}(Ω):\ Ω\subset D,\ |Ω|=m\Big\},$$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$ is either one of the following:\textless{}ul\textgreater{}\textless{}li\textgreater{} the Dirichlet energy $E\_f$, with respect to a (possibly sign-changing) function $f\in L^p$;\textless{}/li\textgreater{}\textless{}li\textgreater{}a spectral functional of the form $F(λ\_{1},\dots,λ\_{k})$, where $λ\_k$ is the $k$th eigenvalue of the Dirichlet Laplacian and $F:\mathbb{R}^k\to\mathbb{R}$ is Lipschitz continuous and increasing in each variable.\textless{}/li\textgreater{}\textless{}/ul\textgreater{}The domain $D$ is the whole space $\mathbb{R}^d$ or a bounded domain. We also give general assumptions on the functional $\mathcal{G}$ so that the result remains valid.
Nonexistence of Minimizers for a Nonlocal Perimeter Functional with a Riesz and a Background Potential
考虑了在体积约束下，含$N\geq2$的$\mathbb{R}^N$中，能量包含非局部周长和一般核$K$，里斯位势和背景势的极小解的不存在性.我们表明，能量有足够大的质量下适当的假设$K$没有极小。证明是基于一个极小的分区和比较的能量的总和为每个部分的原始配置的能量。这一战略见[R. A.弗兰克河Killip，P.T. Nam，2016]和[D.A. La Manna，2018]
