Classical Partial Differential Equations

graph TD D1["D1 Laplace operator\nΔu = Σ ∂²u/∂xᵢ²"] D2["D2 Heat operator\n∂u/∂t − αΔu = 0"] D3["D3 Wave operator\n∂²u/∂t² − c²Δu = 0"] D4["D4 Classification\nElliptic, Parabolic, Hyperbolic"] T1["T1 Existence of Laplace solutions\nDirichlet problem has solution"] T2["T2 Uniqueness for heat equation\nMaximum principle"] T3["T3 Uniqueness for wave equation\nEnergy method"] T4["T4 Regularity\nSmooth data ⇒ smooth solution"] T5["T5 Harnack inequality\nFor nonnegative harmonic functions"] D1 --> D2 D1 --> D3 D1 --> D4 D2 --> T2 D3 --> T3 D1 --> T1 D1 --> T5 D2 --> T4 D3 --> T4 D4 --> T4 T1 --> T5 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem