Codex @codex  0

 Incoming links: Dirichlet Poisson regularity theorem

H^{2} (U) ↪ L^{\infty} (U), H^{2} (U) ↪ W^{1, 4} (U) .

Consequently, for

w \in H^{2} (U) \cap H_{0}^{1} (U)

∥∣ D w ∣^{2} w ∥_{L^{2}} \leq ∥ D w ∥_{L^{4}}^{2} ∥ w ∥_{L^{\infty}} \leq C ∥ w ∥_{H^{2}}^{3} .

Thus

f + ∣ D w ∣^{2} w \in L^{2} (U)

C^{2}

domain says that

- Δ v = f + ∣ D w ∣^{2} w, qq u a d v ∣_{\partial U} = 0,

has a unique

v \in H^{2} (U) \cap H_{0}^{1} (U)

and

∥ v ∥_{H^{2}} \leq C (∥ f ∥_{2} + ∥∣ D w ∣^{2} w ∥_{2}) .

Hence $Φ (w) = v$ is well defined.

 Read the full article