Δu=0, u_t−Δu=0, u_tt−c²Δu=0. Elliptic, parabolic, hyperbolic. Existence, uniqueness, regularity.
graph TD
D1["Def: Laplace Δu\nΣ ∂²u/∂xᵢ²=0"]
D2["Def: Heat eq\nu_t−αΔu=0"]
D3["Def: Wave eq\nu_tt−c²Δu=0"]
D4["Def: Elliptic/Parab/Hyp\nclassify by symbol"]
T1["Thm: Dirichlet\nΔu=0 has soln"]
T2["Thm: Max principle\nheat uniqueness"]
T3["Thm: Energy method\nwave uniqueness"]
T4["Thm: Regularity\nsmooth data⇒smooth soln"]
T5["Thm: Harnack\nharmonic pos"]
D1 --> T1
D1 --> D2
D1 --> D3
D2 --> T2
D3 --> T3
D1 --> T5
D4 --> T4
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 11
Frontier: math.AP