Maximum Principles for PDEs

graph TD D1["D1 Subharmonic function\nΔu ≥ 0 (or Δu ≤ 0)"] D2["D2 Elliptic operator L\nLu = aⁱʲ∂ᵢ∂ⱼu + bⁱ∂ᵢu + cu"] D3["D3 Parabolic operator\n∂u/∂t − Lu = 0"] D4["D4 Comparison principle\nu ≤ v on ∂Ω ⇒ u ≤ v in Ω"] T1["T1 Weak maximum principle\nSubsolution ≤ max of boundary"] T2["T2 Strong maximum principle\nInterior max ⇒ u constant"] T3["T3 Hopf lemma\nNormal derivative at boundary max"] T4["T4 Uniqueness for Dirichlet\nFrom maximum principle"] T5["T5 Harnack inequality\nBounds ratio of positive harmonic functions"] D1 --> T1 D2 --> T1 D2 --> T2 D1 --> T2 D4 --> T4 D2 --> T3 T1 --> T3 T1 --> T4 D1 --> T5 D2 --> T5 T2 --> T4 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