Δu = 0 in Ω, u = g on ∂Ω. Existence/uniqueness for sufficiently regular Ω. Perron's method via subharmonic envelope. Modern approach: Sobolev space + variational. Foundation for boundary-value problems.
Δu = 0 in Ω, u = g on ∂Ω. Existence/uniqueness for sufficiently regular Ω. Perron's method via subharmonic envelope. Modern approach: Sobolev space + variational. Foundation for boundary-value problems.