具有点相互作用和 Coulomb 相互作用的 Schrödinger 算子的 Faber–Krahn 不等式

磁拉普拉斯算子的反向Faber-Krahn不等式

A reverse Faber-Krahn inequality for the magnetic Laplacian

针对带均匀磁场和Neumann边界条件的磁拉普拉斯算子第一特征值，研究反向Faber-Krahn猜想（面积固定时圆盘最大化该特征值），用水平集方法证明磁场足够小时猜想成立。

A Local Faber-Krahn inequality and Applications to Schrödinger's Equation

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Δ+ V$ on an arbitrary domain $Ω$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in Ω$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $Ω$ with probability $\ge 1/2$. For nice (e.g., convex) domains, $T(x_0) \asymp d(x_0,\partialΩ)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $\asymp T(x_0)^{1/2}$ such that $$ \| V \|_{L^{\frac{…

A Faber--Krahn inequality for indented and cut membranes

In 1960, Payne and Weinberger proved that among all domains that lie within a wedge (an angle whose measure is less than or equal to $π$), and have a given value of a certain integral the circular sector has the lowest fundamental eigenvalue of the Dirichlet Laplacian. Here, it is shown that an analogue of this assertion is true for domains with a cut and for indented domains; that is, for those located in a reflex angle (its measure is between $π$ and $2 π$).

基于随机游走展开的量子图热含量Faber-Krahn不等式

Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion

该研究探究量子图热含量的Rayleigh-Faber-Krahn不等式类似结论，用谱理论和随机游走方法证明其在大小时间极端情形成立，还提出热含量的新随机游走表达式，全时段结论仍待解决。

带负边界参数的Robin拉普拉斯算子的反Faber-Krahn不等式：所有维数下的小耦合

A reverse Faber--Krahn inequality for the Robin Laplacian with negative boundary parameter: small coupling in all dimensions

针对凸域在弱边界耦合条件下，证明了Bareket猜想：当边界参数足够接近零时，球体在固定体积的有界凸域中最大化第一特征值。

A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegö type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.