garywelz/programming_framework
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1<!DOCTYPE html>2<html lang="en">3<head>4 <meta charset="UTF-8">5 <meta name="viewport" content="width=device-width, initial-scale=1.0">6 <title>Maximum Principles for PDEs - Mathematics Process</title>7 <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>8 <style>9 * { margin: 0; padding: 0; box-sizing: border-box; }10 body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); min-height: 100vh; padding: 20px; }11 .container { max-width: 1200px; margin: 0 auto; background: white; border-radius: 15px; box-shadow: 0 20px 40px rgba(0,0,0,0.1); overflow: hidden; }12 .header { background: #2980b9; color: white; padding: 30px; }13 .header h1 { margin: 0 0 10px 0; font-size: 2em; font-weight: 300; }14 .header-meta { display: flex; flex-wrap: wrap; gap: 15px; margin-top: 15px; font-size: 0.9em; opacity: 0.9; }15 .meta-item { background: rgba(255,255,255,0.2); padding: 5px 12px; border-radius: 20px; }16 .nav-links { padding: 15px 30px; background: #f8f9fa; border-bottom: 1px solid #ecf0f1; }17 .nav-links a { color: #2980b9; text-decoration: none; margin-right: 20px; font-weight: 500; }18 .nav-links a:hover { text-decoration: underline; }19 .content { padding: 30px; }20 .description { margin-bottom: 30px; }21 .flowchart-container { margin: 30px 0; }22 .flowchart-container h2 { color: #2c3e50; margin-bottom: 15px; }23 .mermaid { background: white; padding: 20px; border-radius: 10px; border: 1px solid #ecf0f1; overflow-x: auto; }24 .color-legend { background: #f8f9fa; padding: 20px; border-radius: 10px; margin: 30px 0; }25 .color-legend h3 { color: #2c3e50; margin-bottom: 15px; }26 .color-grid { display: grid; grid-template-columns: repeat(auto-fit, minmax(180px, 1fr)); gap: 15px; }27 .color-item { display: flex; align-items: center; gap: 10px; padding: 10px; background: white; border-radius: 5px; }28 .color-box { width: 30px; height: 30px; border-radius: 4px; border: 1px solid #ddd; }29 .info-card { background: #f8f9fa; padding: 20px; border-radius: 10px; margin-top: 20px; }30 .info-card h3 { color: #2c3e50; margin-bottom: 15px; }31 .info-card ul { list-style: none; padding: 0; }32 .info-card li { padding: 8px 0; border-bottom: 1px solid #ecf0f1; }33 .info-card li:last-child { border-bottom: none; }34 </style>35</head>36<body>37 <div class="container">38 <div class="header">39 <h1>Maximum Principles for PDEs</h1>40 <div class="header-meta">41 <span class="meta-item">Partial Differential Equations</span>42 <span class="meta-item">Mathematics</span>43 </div>44 </div>45 <div class="nav-links">46 <a href="../../mathematics_index.html">← Back to Mathematics Index</a>47 <a href="partial_differential_equations-index.html">Partial Differential Equations Index</a>48 <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv: Analysis of PDEs (math.AP)</a>49 </div>50 <div class="content">51 <div class="description">52 <h2>Description</h2>53 <p>Maximum principles for elliptic and parabolic equations. Harmonic functions achieve extrema on the boundary; heat equation preserves maximum location.</p>54 </div>55 <div class="flowchart-container">56 <h2>Dependency Flowchart</h2>57 <div class="mermaid">58graph TD59 D1["D1 Subharmonic function\nΔu ≥ 0 (or Δu ≤ 0)"]60 D2["D2 Elliptic operator L\nLu = aⁱʲ∂ᵢ∂ⱼu + bⁱ∂ᵢu + cu"]61 D3["D3 Parabolic operator\n∂u/∂t − Lu = 0"]62 D4["D4 Comparison principle\nu ≤ v on ∂Ω ⇒ u ≤ v in Ω"]63 64 T1["T1 Weak maximum principle\nSubsolution ≤ max of boundary"]65 T2["T2 Strong maximum principle\nInterior max ⇒ u constant"]66 T3["T3 Hopf lemma\nNormal derivative at boundary max"]67 T4["T4 Uniqueness for Dirichlet\nFrom maximum principle"]68 T5["T5 Harnack inequality\nBounds ratio of positive harmonic functions"]69 70 D1 --> T171 D2 --> T172 D2 --> T273 D1 --> T274 D4 --> T475 D2 --> T376 T1 --> T377 T1 --> T478 D1 --> T579 D2 --> T580 T2 --> T481 82 classDef definition fill:#3498db,color:#fff,stroke:#2980b983 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a08584 class D1,D2,D3,D4 definition85 class T1,T2,T3,T4,T5 theorem86 </div>87 </div>88 <div class="color-legend">89 <h3>Color Scheme</h3>90 <div class="color-grid">91 <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div>92 <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div>93 </div>94 </div>95 <div class="info-card">96 <h3>Info</h3>97 <ul>98 <li><strong>Subcategory:</strong> partial_differential_equations</li>99 <li><strong>Keywords:</strong> maximum principle, Hopf lemma, Harnack, harmonic, elliptic</li>100 <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv math.AP</a></li>101 </ul>102 </div>103 </div>104 </div>105 <script>106 mermaid.initialize({ startOnLoad: true, theme: 'default', flowchart: { useMaxWidth: true, htmlLabels: true, curve: 'step', nodeSpacing: 25, rankSpacing: 40 } });107 </script>108</body>109</html>110