39.2. AN APPLICATION OF THE MOUNTAIN PASS THEOREM 1335

Corollary 39.1.6 Let U be a connected open set with the property that for some sequence,ηk ↓ 0,

U = ∪∞k=1Uηk

for Uηk a connected open set and suppose u ∈W 1,p (U) and ∇u = 0 a.e. Then u equals aconstant a.e.

Example 39.1.7 Let U be a bounded open connected subset of Rn and let V be a closedsubspace of H1 (U) defined by

V ≡{

u ∈ H1 (U) : γu = 0 on Γ}

where the surface measure of Γ is positive.Let α i j ∈ L∞ (U) for i, j = 1,2, · · · ,n and define A : V →V ′ by

A(u)(v)≡∫

Uα

i j (x)u,i (x)v, j (x)dx.

forα

i jviv j ≥ δ |v|2

whenever v ∈ Rn. Then A maps V to V ′ one to one and onto.

This follows from Theorem 39.1.5 using the equivalent norm defined there. DefineF ∈V ′ by ∫

Uf (x)v(x)dx+

∫∂U\Γ

g(x)γv(x)dx

for f ∈ L2 (U) and g ∈ H1/2 (∂U) . Then the equation,

Au = F in V ′

which is equivalent to u ∈V and for all v ∈V,∫U

αi j (x)u,i (x)v, j (x)dx =

∫U

f (x)v(x)dx+∫

∂U\Γg(x)γv(x)dµ

is a weak solution for the boundary value problem,

−(α

i ju,i), j = f in U, α

i ju,in j = g on ∂U \Γ, u = 0 on Γ

as you can verify by using the divergence theorem formally.

39.2 An Application Of The Mountain Pass TheoremRecall the mountain pass theorem 24.1.3.

Theorem 39.2.1 Let H be a Hilbert space and let I : H → R be a C1 functional having I′

Lipschitz continuous and such that I satisfies the Palais Smale condition. Suppose I (0) = 0