The Monotonicity of the Principal Frequency of the Anisotropic pp-Laplacian

Let $D>1$ be a fixed integer. Given a smooth bounded, convex domain $\Omega \subset \mathbb{R}^D$ and $H:\mathbb{R}^D\rightarrow [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }(\mathbb{R}^D\setminus \lbrace 0\rbrace )$ for which the Hessian matrix $D^2(H^p)$ is positive definite in $\mathbb{R}^D\setminus \lbrace 0\rbrace$ for any $p\in (1,\infty )$, we study the monotonicity of the principal frequency of the anisotropic $p$-Laplacian (constructed using the function $H$) on $\Omega$ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega$ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $\mathbb{R}^D$, we recover some recent results obtained by the first two authors in [3, 4]

The Monotonicity of the Principal Frequency of the Anisotropic pp-Laplacian

Let $D>1$ be a fixed integer. Given a smooth bounded, convex domain $\Omega \subset \mathbb{R}^D$ and $H:\mathbb{R}^D\rightarrow [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }(\mathbb{R}^D\setminus \lbrace 0\rbrace )$ for which the Hessian matrix $D^2(H^p)$ is positive definite in $\mathbb{R}^D\setminus \lbrace 0\rbrace$ for any $p\in (1,\infty )$, we study the monotonicity of the principal frequency of the anisotropic $p$-Laplacian (constructed using the function $H$) on $\Omega$ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega$ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $\mathbb{R}^D$, we recover some recent results obtained by the first two authors in [3, 4]

Equi-integrability results for 3D-2D dimension reduction problems

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients ( ∇αuε ∣ 1 bounded in L p (Ω; R 9), 1 < p < +∞. Here it is shown that, up to a subsequence, uε may be decomposed as wε +zε, where zε carries all the concentration effects, i.e. { ∣ ∣ ( ∇αwε | 1 ε ∇3wε ∣p} is equi-integrable, and wε captures the oscillatory behavior, i.e. zε → 0 in measure. In addition, if {uε} is a recovering sequence then zε = zε(xα) nearby ∂Ω