Suppose that Ω ⊆ R n is bounded, and path-connected, and u ∈ C2 (Ω) ∩...

Suppose that Ω ⊆ R n is bounded, and path-connected, and u ∈ C² (Ω) ∩ C(∂Ω) satisfies ( −∆u = 0 in Ω, u = g on ∂Ω. Prove that if g ∈ C(∂Ω) with g(x) = ( ≥ 0 for all x ∈ ∂Ω, > 0 for some x ∈ ∂Ω, then u(x) > 0 for all x ∈ Ω

Expert Solution

Here we are using strong maximum principle of harmonic function which is applicable in our case as the domain is given to be path-connected.

Colby Messinger answered 2 years ago

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