论文标题
从寄生虫不平等到仿射光谱不平等现象
From affine Poincaré inequalities to affine spectral inequalities
论文作者
论文摘要
给定一个有限的开放子集$ω$的$ \ mathbb r^n $，我们确定了仿射球$ b^{\ mathcal a} _p（ω）= \ {f \ in w^{1，p} _0（ω）：\ \ \ \ \ \ \ \ \ \ leq 1 \ \ the \ use f \ use f \ use f \ use f \ use f \ use f \ use f \ use f \ use f \ pation faction， E_PF $由Lutwak，Yang和Zhang在[43]中以及其紧凑性以及任何$ p \ geq 1 $中的$ l^p（ω）$中。这些观点强烈使用了著名的Blaschke-Santaló不平等。作为对应的，我们在有限域中开发了$ p $ -rayleigh商的基本理论，在仿射案例中，以$ p \ geq 1 $。更具体地说，我们建立了$ p $ - 携带版本的庞加莱不平等现象及其后果。我们介绍了仿射不变$ p $ -laplace运算符$Δ_p^{\ Mathcal a} f $定义了$ p $ - affine rayleigh商的最小化问题的欧拉 - 拉格朗日方程。我们还研究了它的第一个特征值$λ^{\ Mathcal a} _ {1，p}（ω）$，它满足相应的仿期faber-krahn不平等，这是$λ^{\ mathcal a} _} _ {1，p}（1，p}（ω）$ s nim see n e el ell e ell ell ell ell ell se n ell ell seles in ell ell se nip se nip sebless $。这一点从根本上取决于针对操作员$Δ_P^{\ MATHCAL A} F $的PDES规则性分析。我们还提供了仿射和经典特征值之间的一些比较，包括通过表征$ p \ geq 1 $的平等案例的刚性结果。获得的所有仿射不平等都更强，直接暗示了经典的不平等。
Given a bounded open subset $Ω$ of $\mathbb R^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(Ω) = \{f \in W^{1,p}_0(Ω):\ \mathcal E_p f \leq 1\}$ with respect to the affine functional $\mathcal E_pf$ introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in $L^p(Ω)$ for any $p \geq 1$. These points use strongly the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p\geq 1$. More specifically, we establish $p$-affine versions of the Poincaré inequality and some of their consequences. We introduce the affine invariant $p$-Laplace operator $Δ_p^{\mathcal A} f$ defining the Euler-Lagrange equation of the minimization problem of the $p$-affine Rayleigh quotient. We also study its first eigenvalue $λ^{\mathcal A}_{1,p}(Ω)$ which satisfies the corresponding affine Faber-Krahn inequality, this is that $λ^{\mathcal A}_{1,p}(Ω)$ is minimized (among sets of equal volume) only when $Ω$ is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator $Δ_p^{\mathcal A} f$. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for $p \geq 1$. All affine inequalities obtained are stronger and directly imply the classical ones.
