论文标题
定量形式的退化自由不连续性问题和光谱不平等
Degenerate free discontinuity problems and spectral inequalities in quantitative form
论文作者
论文摘要
我们介绍了一种新的几何分析功能，我们在自由不连续问题的背景下进行了分析。它的主要特征是几何术语（跳跃集的长度）出现为负符号。这是通过搜索定量不等式的索波列夫 - 帕卡雷不平等常数，其痕量术语的最佳常数为$ \ mathbb {r}^n $，这对应于与laplace ocerator的半连续性问题相关的基本特征值。我们的方法基于对这项新的，退化的功能的研究，该功能涉及与跳跃集合相互作用的障碍问题。最终，这将变成一个混合的自由不连续性/自由边界问题，分别在障碍级别上方/处发生。
We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with negative sign. This is motivated by searching quantitative inequalities for best constants of Sobolev-Poincaré inequalities with trace terms in $\mathbb{R}^n$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.
