Sobolev Spaces and Weak Solutions

graph TD D1["D1 Sobolev space W^{k,p}\nFunctions with k weak derivatives in L^p"] D2["D2 Weak derivative\nDistributional derivative"] D3["D3 Weak solution\nSatisfies PDE in integral sense"] D4["D4 H¹ and H₀¹\nHilbert Sobolev spaces"] T1["T1 Sobolev embedding\nW^{k,p} ⊂ L^q for p ≤ q*"] T2["T2 Rellich-Kondrachov\nW^{1,p} ⋐ L^p compact"] T3["T3 Lax-Milgram for weak form\nUnique weak solution exists"] T4["T4 Regularity: f ∈ L² ⇒ u ∈ H²\nFor elliptic with smooth coeffs"] T5["T5 Caccioppoli inequality\nInterior gradient bounds"] D1 --> T1 D2 --> D1 D3 --> T3 D4 --> T3 D1 --> T2 D1 --> T4 D3 --> T4 D1 --> T5 T1 --> T4 T2 --> T3 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