Hadamard: existence, uniqueness, continuous dependence. Cauchy–Kowalevski. Energy estimates. Lax–Milgram.
graph TD
D1["Def: Well-posed\n exist, unique, cont dep"]
D2["Def: Cauchy problem\ninit data on surface"]
D3["Def: Ill-posed\nviolates Hadamard"]
D4["Def: Energy estimate\n‖u‖ ≤ C‖data‖"]
T1["Thm: Cauchy–Kowalevski\nanalytic ⇒ analytic soln"]
T2["Thm: Hadamard ex\nLaplace Cauchy ill-posed"]
T3["Thm: Energy est\nparab/hyp stability"]
T4["Thm: Lax–Milgram\nelliptic in Hilbert"]
T5["Thm: Semigroup\n evolution well-posed"]
D1 --> T4
D2 --> T1
D3 --> T2
D4 --> T3
T3 --> D1
T4 --> D1
T5 --> D1
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