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Code for: The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II

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Auteurs principaux:	Endo, Ryoki, Liu, Xuefeng
Format:	Recurso digital
Publié:	Zenodo 2025
Sujets:	spectral geometry computer-assisted proof Dirichlet eigenvalues interval arithmetic MATLAB
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_version_	1866901793746714624
author	Endo, Ryoki Liu, Xuefeng
author_facet	Endo, Ryoki Liu, Xuefeng
contents	<p>This project provides the source code and computational framework for the computer-assisted proof presented in the paper:</p> <blockquote> <p><strong>[Paper B] The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles</strong> https://arxiv.org/abs/2305.14063 <em>(to appear in Numerische Mathematik)</em></p> </blockquote> <p>The primary goal is to rigorously validate the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to a conjecture posed by R. Laugesen and B. Siudeja.</p> <p>The computer-assisted proof for nearly degenerate triangles is presented in the following paper:</p> <blockquote> <p><strong>[Paper A] The second Dirichlet eigenvalue is simple on every non-equilateral triangle, Part I: Nearly degenerate triangles</strong> <em>Journal of Differential Equations 447, 113629</em> https://doi.org/10.1016/j.jde.2025.113629</p> </blockquote> <h2>Background</h2> <p>Determining the eigenvalue multiplicity of the Laplace operator is challenging, especially when eigenvalues are nearly degenerate—as for the second and third Dirichlet eigenvalues on equilateral triangles where λ2 = λ3. Standard numerical methods struggle to separate these tightly clustered eigenvalues with mathematical rigor.</p> <p>We split the parameter space of triangles Ω into three regions:</p> <ul> <li><strong>Region Ω_up</strong> (nearly equilateral triangles): Using <strong>Algorithm 1</strong>, we prove the difference quotient satisfies D(λ2) < D(λ3), implying separation.</li> <li><strong>Region Ω_down</strong>: Using <strong>Algorithm 2</strong>, we compute high-precision eigenvalue bounds directly to show (λ2 <=) sup(λ2) < inf(λ3) (<= λ3).</li> <li><strong>Region Ω_rest</strong> (nearly degenerate triangles): This case is handled in <strong>[Paper A]</strong>.</li> </ul> <p>For the definition of each region, see [Paper B].</p> <h2>Core Libraries & Dependencies</h2> <p>This project relies on specialized libraries for verified numerical computation:</p> <ol> <li><strong>INTLAB</strong>: The fundamental toolbox for rigorous interval arithmetic in MATLAB.<ul> <li>Source: http://www.tuhh.de/ti3/intlab/ [INTLAB_V12, INTLAB_V14 were used.]</li> </ul> </li> <li>Revised version of <strong>VFEM2D</strong>: Used for rigorous finite element matrix assembly and high-precision eigenvalue bounds (Lehmann–Goerisch method).<ul> <li>Source: https://github.com/xfliu/VFEM2D [2025/12/13]</li> </ul> </li> <li><strong>veigs</strong>: Used for solving generalized matrix eigenvalue problems with rigorous error bounds.<ul> <li>Source: https://github.com/yuuka-math/veigs [2025/12/13]</li> </ul> </li> </ol> <h2>Installation & Usage</h2> <p>Before running the code, ensure you have <strong>MATLAB</strong> (R2020b or later), <strong>INTLAB</strong>, and <strong>Gmsh</strong>.</p> <h3>Setup</h3> <p>Clone the repository and configure <code>my_env_config.m</code> to set the paths for INTLAB and Gmsh.</p> <pre><code>% Open my_env_config.m % Switch for approximate computation mode and rigorous computation mode global INTERVAL_MODE; INTERVAL_MODE=1; % The computation was tested with gmsh version 4.8.4. %For mesh generated by other versions of gmsh, the results may vary slightly. global gmsh_command gmsh_command = '/path/to/gmsh' %Path of gmsh command.; global mesh_path mesh_path = '/tmp/' %Place to save temporary mesh files. addpath('/path/to/your/INTLAB_directory'); % e.g., addpath('/Applications/Intlab_V12'); </code></pre> <h3>Execution</h3> <p>The <code>ProofRunner</code> class is the central controller.</p> <ol> <li><p><strong>Initialization:</strong></p> <p>Matlab</p> <pre><code>s = ProofRunner; s.setupAll(); </code></pre> </li> <li><p><strong>Algorithm 1 (Region Ω_up):</strong></p> <p>Matlab</p> <pre><code>s.runAlgo1All(); % Sweeps the angular range delta in [0, pi/3] </code></pre> </li> <li><p><strong>Algorithm 2 (Region Ω_down):</strong></p> <p>Matlab</p> <pre><code>s.runAlgo2All(); % Batch verification using inputs/cell_def.csv </code></pre> </li> </ol> <p><em>For detailed instructions, please refer to the <code>README.md</code> file included in this repository.</em></p>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_17929868
institution	Zenodo
language
publishDate	2025
publisher	Zenodo
record_format	zenodo
spellingShingle	Code for: The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II Endo, Ryoki Liu, Xuefeng spectral geometry computer-assisted proof Dirichlet eigenvalues interval arithmetic MATLAB <p>This project provides the source code and computational framework for the computer-assisted proof presented in the paper:</p> <blockquote> <p><strong>[Paper B] The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles</strong> https://arxiv.org/abs/2305.14063 <em>(to appear in Numerische Mathematik)</em></p> </blockquote> <p>The primary goal is to rigorously validate the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to a conjecture posed by R. Laugesen and B. Siudeja.</p> <p>The computer-assisted proof for nearly degenerate triangles is presented in the following paper:</p> <blockquote> <p><strong>[Paper A] The second Dirichlet eigenvalue is simple on every non-equilateral triangle, Part I: Nearly degenerate triangles</strong> <em>Journal of Differential Equations 447, 113629</em> https://doi.org/10.1016/j.jde.2025.113629</p> </blockquote> <h2>Background</h2> <p>Determining the eigenvalue multiplicity of the Laplace operator is challenging, especially when eigenvalues are nearly degenerate—as for the second and third Dirichlet eigenvalues on equilateral triangles where λ2 = λ3. Standard numerical methods struggle to separate these tightly clustered eigenvalues with mathematical rigor.</p> <p>We split the parameter space of triangles Ω into three regions:</p> <ul> <li><strong>Region Ω_up</strong> (nearly equilateral triangles): Using <strong>Algorithm 1</strong>, we prove the difference quotient satisfies D(λ2) < D(λ3), implying separation.</li> <li><strong>Region Ω_down</strong>: Using <strong>Algorithm 2</strong>, we compute high-precision eigenvalue bounds directly to show (λ2 <=) sup(λ2) < inf(λ3) (<= λ3).</li> <li><strong>Region Ω_rest</strong> (nearly degenerate triangles): This case is handled in <strong>[Paper A]</strong>.</li> </ul> <p>For the definition of each region, see [Paper B].</p> <h2>Core Libraries & Dependencies</h2> <p>This project relies on specialized libraries for verified numerical computation:</p> <ol> <li><strong>INTLAB</strong>: The fundamental toolbox for rigorous interval arithmetic in MATLAB.<ul> <li>Source: http://www.tuhh.de/ti3/intlab/ [INTLAB_V12, INTLAB_V14 were used.]</li> </ul> </li> <li>Revised version of <strong>VFEM2D</strong>: Used for rigorous finite element matrix assembly and high-precision eigenvalue bounds (Lehmann–Goerisch method).<ul> <li>Source: https://github.com/xfliu/VFEM2D [2025/12/13]</li> </ul> </li> <li><strong>veigs</strong>: Used for solving generalized matrix eigenvalue problems with rigorous error bounds.<ul> <li>Source: https://github.com/yuuka-math/veigs [2025/12/13]</li> </ul> </li> </ol> <h2>Installation & Usage</h2> <p>Before running the code, ensure you have <strong>MATLAB</strong> (R2020b or later), <strong>INTLAB</strong>, and <strong>Gmsh</strong>.</p> <h3>Setup</h3> <p>Clone the repository and configure <code>my_env_config.m</code> to set the paths for INTLAB and Gmsh.</p> <pre><code>% Open my_env_config.m % Switch for approximate computation mode and rigorous computation mode global INTERVAL_MODE; INTERVAL_MODE=1; % The computation was tested with gmsh version 4.8.4. %For mesh generated by other versions of gmsh, the results may vary slightly. global gmsh_command gmsh_command = '/path/to/gmsh' %Path of gmsh command.; global mesh_path mesh_path = '/tmp/' %Place to save temporary mesh files. addpath('/path/to/your/INTLAB_directory'); % e.g., addpath('/Applications/Intlab_V12'); </code></pre> <h3>Execution</h3> <p>The <code>ProofRunner</code> class is the central controller.</p> <ol> <li><p><strong>Initialization:</strong></p> <p>Matlab</p> <pre><code>s = ProofRunner; s.setupAll(); </code></pre> </li> <li><p><strong>Algorithm 1 (Region Ω_up):</strong></p> <p>Matlab</p> <pre><code>s.runAlgo1All(); % Sweeps the angular range delta in [0, pi/3] </code></pre> </li> <li><p><strong>Algorithm 2 (Region Ω_down):</strong></p> <p>Matlab</p> <pre><code>s.runAlgo2All(); % Batch verification using inputs/cell_def.csv </code></pre> </li> </ol> <p><em>For detailed instructions, please refer to the <code>README.md</code> file included in this repository.</em></p>
title	Code for: The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II
topic	spectral geometry computer-assisted proof Dirichlet eigenvalues interval arithmetic MATLAB
url	https://doi.org/10.5281/zenodo.17929868