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Sharp estimates for fundamental frequencies of elliptic operators generated by asymmetric seminorms in low dimensions

Julian Haddad, Raul Fernandes Horta, Marcos Montenegro

TL;DR

The paper develops sharp asymmetries in Poincaré-type inequalities in 1D and transfers the insights to a comprehensive 2D theory for fundamental frequencies of nonlinear elliptic operators generated by asymmetric seminorms. It provides explicit constants and extremizers in 1D, classifies nondegenerate and degenerate 2D anisotropies, and connects spectral bounds to optimal anisotropic design via width functions. A key outcome is the identification of rigid upper bounds attained by the isotropic $p$-Laplacian and a precise description of lower bounds through geometric widths, together with an isoperimetric conjecture for 2D anisotropic operators. The results establish foundational links between geometry, anisotropy, and spectral optimization for nonlinear elliptic operators in low dimensions, with implications for optimal design and eigenvalue control.

Abstract

We establish new sharp asymmetric Poincare inequalities in one-dimension with the computation of optimal constants and characterization of extremizers. Using the one-dimensional theory we develop a comprehensive study on fundamental frequencies in the plane and related spectral optimization in the very general setting of positively homogeneous anisotropies.

Sharp estimates for fundamental frequencies of elliptic operators generated by asymmetric seminorms in low dimensions

TL;DR

The paper develops sharp asymmetries in Poincaré-type inequalities in 1D and transfers the insights to a comprehensive 2D theory for fundamental frequencies of nonlinear elliptic operators generated by asymmetric seminorms. It provides explicit constants and extremizers in 1D, classifies nondegenerate and degenerate 2D anisotropies, and connects spectral bounds to optimal anisotropic design via width functions. A key outcome is the identification of rigid upper bounds attained by the isotropic -Laplacian and a precise description of lower bounds through geometric widths, together with an isoperimetric conjecture for 2D anisotropic operators. The results establish foundational links between geometry, anisotropy, and spectral optimization for nonlinear elliptic operators in low dimensions, with implications for optimal design and eigenvalue control.

Abstract

We establish new sharp asymmetric Poincare inequalities in one-dimension with the computation of optimal constants and characterization of extremizers. Using the one-dimensional theory we develop a comprehensive study on fundamental frequencies in the plane and related spectral optimization in the very general setting of positively homogeneous anisotropies.
Paper Structure (8 sections, 21 theorems, 190 equations, 1 figure)

This paper contains 8 sections, 21 theorems, 190 equations, 1 figure.

Table of Contents

  1. Overview and main statements
  2. Asymmetric Poincaré inequalities for $n = 1$
  3. The case $b = 0$
  4. The case $b > 0$
  5. Fundamental frequencies for $n = 2$: the non-degenerate case
  6. Fundamental frequencies for $n = 2$: degenerate cases
  7. Extended maximization conjecture for $n = 2$
  8. Anisotropic spectral optimization for $n = 2$

Key Result

Theorem 1.1

Let $p > 1$ and let $T > 0$. For any $a > 0$ and $b \geq 0$, we have: Consequently, since $u \in W_0^{1,p}(-T,T)$ only when $b > 0$, $\lambda_{p}^{a,b}(-T,T)$ is a fundamental frequency if and only if $b > 0$. In other words,

Figures (1)

  • Figure 1: The functions $u_p$ for $p=2$ and the values $b=1-a$ and $a = 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3: The case $b=0$
  • Theorem 1.4: The case $b>0$
  • Corollary 1.2
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • ...and 28 more