标题： 关于具有竞争非局部项的等周问题。 II. 一般情况 显示英文标题

标题： On an isoperimetric problem with a competing non-local term. II. The general case

摘要： 本文是[H. Knüpfer和C. B. Muratov, Commun. Pure Appl. Math. (2012, 待发表)]的延续。我们研究了经典的等周问题，该问题通过添加一个由距离的反幂次核产生的非局部排斥项进行了修改。在这项工作中，我们处理了一般空间维度的情况。我们获得了具有足够小质量的极小值解的基本存在结果。对于核中指数的某些范围，我们也获得了足够大质量下的不存在结果，以及在足够小质量和低空间维度下极小值解作为球体的特征。三个空间维度和库仑排斥的物理上重要的特殊情况包含在上述所有结果中。特别是，我们的工作给出了对经典液滴模型框架下任意高原子序数的稳定原子核是否存在这一问题的否定答案。在所有情况下，对于大质量，最小能量与质量成线性比例，即使能量的下确界可能未被达到。 显示英文摘要

摘要： This paper is the continuation of [H. Kn\"upfer and C. B. Muratov, Commun. Pure Appl. Math. (2012, to be published)]. We investigate the classical isoperimetric problem modified by an addition of a non-local repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel we also obtain non-existence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy may not be attained.