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Spectral bounds for the operator pencil of an elliptic system in an angle

Michael Tsopanopoulos

TL;DR

This work develops a spectral-pencil framework for a planar-angle model problem of linear elliptic systems under Dirichlet, mixed, and Neumann boundary conditions. By factoring the leading symbol and employing a monic reduction with a standard root $V$, the authors reduce boundary-value solvability to spectral questions for a parameter-dependent matrix pencil $\mathcal{A}(\lambda)$ and associated matrices $M_{\lambda,\alpha}$. They establish optimal lower bounds on $|\Re\lambda|$ for Dirichlet and, centrally, mixed boundary conditions, under mild ellipticity and (for mixed) contractive Neumann well-posedness assumptions; these bounds translate into precise regularity results in two dimensions (and edge-induced reductions in 3D). Neumann analysis remains more delicate, with results tied to complementing boundary conditions and accretive-operator theory. The bounds are shown to be sharp via explicit constructions, and the scalar case is discussed as a clarifying special case.

Abstract

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Spectral bounds for the operator pencil of an elliptic system in an angle

TL;DR

This work develops a spectral-pencil framework for a planar-angle model problem of linear elliptic systems under Dirichlet, mixed, and Neumann boundary conditions. By factoring the leading symbol and employing a monic reduction with a standard root , the authors reduce boundary-value solvability to spectral questions for a parameter-dependent matrix pencil and associated matrices . They establish optimal lower bounds on for Dirichlet and, centrally, mixed boundary conditions, under mild ellipticity and (for mixed) contractive Neumann well-posedness assumptions; these bounds translate into precise regularity results in two dimensions (and edge-induced reductions in 3D). Neumann analysis remains more delicate, with results tied to complementing boundary conditions and accretive-operator theory. The bounds are shown to be sharp via explicit constructions, and the scalar case is discussed as a clarifying special case.

Abstract

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Paper Structure

This paper contains 28 sections, 27 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and prerequisites
  3. The model problem
  4. Ellipticity
  5. The operator pencil
  6. The case $\lambda=0$
  7. Analysis of the model problem without boundary conditions
  8. Algebraic properties of $L_A$
  9. Solutions for the model problem without boundary conditions
  10. Analysis of the model problem with boundary conditions
  11. Dirichlet boundary conditions
  12. Mixed boundary conditions
  13. Neumann boundary conditions
  14. Investigation of the spectrum of $M_{\lambda,\alpha}$
  15. Numerical range
  16. ...and 13 more sections

Key Result

Lemma 3.2

Assume that $A_{11},A_{12},A_{22}\in \operatorname{Mat}_\ell(\mathbb{R})$ are symmetric matrices and $A_{11}$, $A_{22}$ are positive definite. Then the following are equivalent: In these cases, there exists the factorization for some $V\in \operatorname{Mat}_\ell(\mathbb{C})$ with $\sigma(V)\subset \operatorname{UHP}$.

Figures (2)

  • Figure 1: Relation between $\Re \lambda$ and $\alpha\in [1,2\pi]$ for different boundary conditions. The elliptic tuple is defined by $A_{11}=50.60.61.5$, $A_{12}=0.25-0.4-0.4-0.2$, $A_{22}=1001$. The branches for Dirichlet and Neumann boundary conditions are very close to each other.
  • Figure 2: Left: Relation between $\Re \lambda$ and $\alpha\in [\pi,2\pi]$ for Neumann boundary conditions. The standard root of the monic elliptic tuple is given by $V=(S+i\operatorname{Id}_\ell)D$ for $S=0002$, $D=2112$. Although this tuple is contractive Neumann well-posed, it is not formal positive. The qualitative behavior differs from the Dirichlet and mixed case: Branch merging and $|\Re \lambda|<1/2$ occurs. Only selected branches are plotted. Right: Relation between $\Re \lambda$ and $\alpha\in [1,2\pi]$ for different boundary conditions. The (scalar) elliptic tuple is defined by the standard root $V=-10+i$. In this case, the single branch for Neumann and Dirichlet boundary conditions coincides, and all branches approximate the bounds given in Theorem \ref{['main dir']} and \ref{['main mixed']}.

Theorems & Definitions (58)

  • Definition 3.1
  • Remark
  • Lemma 3.2
  • proof
  • Example 3.3
  • Remark : Fundamental solutions for $\mathcal{L}_A(\partial_\varphi,\lambda)v=0$
  • Example 3.4: Laplace equation
  • Lemma 3.5
  • proof
  • Lemma 4.1
  • ...and 48 more