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 a negative sign. This is motivated by searching quantitative inequalities for the 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.

我们引入了一种新的几何解析泛函，我们可以在自由不连续问题的背景下对其进行分析。它的主要特点是几何项（跳跃集的长度）以负号出现。这是通过在\({\mathbb {R}}^n\)跟踪项中搜索 Sobolev–Poincaré 不等式的最佳常数的定量不等式来推动的，这些项对应于与具有 Robin 边界条件的拉普拉斯算子的半线性问题相关的基本特征值. 我们的方法基于对这种新的退化函数的研究，该函数涉及与跳跃集交互的障碍问题。最终，这成为分别发生在障碍物上方/水平处的混合自由不连续性/自由边界问题。