Define
S = {y ∈ C1[0, π] : у(0) = у(π) = 0}
\(\rm \|f\|_{\infty}=\max _{x \displaystyle \in[0, \pi]}|f(x)|\), for all f ∈ S
B0(f, ε) = {f ∈ S : ||f||∞ < ε}
B1(f, ε) = {f ∈ S : ||f||∞ + ||f'||∞ < ε}
Consider the functional J : S → ℝ given by
J[y] = \(\rm \int_0^\pi(1-\left(y^{\prime})^2\right) y^2 d x\)
Then there exists ε > 0 such that
1
J[y] ≤ J[0], for all y ∈ B0(0, ε)
2
J[y] ≤ J[0], for all y ∈ B1(0, ε)
3
J[y] ≥ J[0], for all y ∈ B0(0, ε)
4
J[y] ≥ J[0], for all y ∈ B1(0, ε)