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A potential well theory for the wave equation with nonlinear by Vitillaro E.

By Vitillaro E.

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Additional resources for A potential well theory for the wave equation with nonlinear source and boundary damping terms

Example text

Let be open and bounded. e. imply λ1 (− + p) ≤ λ1 (− + q); (b) λ1 is a continuous function of q, in the sense that |λ1 (− + q) − λ1 (− + p)| ≤ q − p 8. Let by be open and bounded. For I (u) = 1 2 h ∈ L2 ( |∇u|2 dx − λ 2 L∞ ( ) . ) and λ ∈ R, define I : H01 ( ) → R u2 dx − Prove that I is coercive on H01 ( ) if and only if λ < λ1 . hu dx. 9 Bibliographical Notes 37 9. More generally, assume that and h are in the previous exercise, let q ∈ L∞ ( ), and define J : H01 ( ) → R by J (u) = 1 2 |∇u|2 dx + 1 2 q(x)u2 dx − hu dx.

15) is true. 3 We first show that u∗ + v∗ is a critical point for I . For every u ∈ X1 and every t ∈ R, let γ (t) = J (u∗ + tu, v∗ ) = I (u∗ + tu + v∗ ). The function γ is differentiable and has a maximum point at t = 0, by the preceding lemma. Then 0 = γ (0) = I (u∗ + v∗ )u. In the same way, if for v ∈ X2 we define η(t) = J (u∗ , v∗ + tv) = I (u∗ + v∗ + tv), then we see that η is differentiable and has a minimum point at t = 0. So, 0 = η (0) = I (u∗ + v∗ )v. Adding the two equations we obtain I (u∗ + v∗ )(u + v) = 0 for every u ∈ X1 and every v ∈ X2 ; since H01 ( ) = X1 ⊕ X2 , we conclude that I (u∗ + v∗ ) = 0.

Then I has a global minimum point. Proof Define m = infu∈X I (u). Let {uk }k∈N ⊂ X be a minimizing sequence. Coercivity implies that {uk }k∈N is bounded. Since X is reflexive, by the Banach–Alaoglu Theorem we can extract from {uk }k∈N a subsequence, still denoted uk , such that uk converges weakly to some u ∈ X. 3 we then obtain I (u) ≤ lim inf I (uk ) = m. k→∞ Therefore I (u) = m and u is a global minimum for I . 3). A more general statement is thus the following version of the Weierstrass Theorem: let X be a reflexive Banach space and let I : X → R be weakly lower semicontinuous and coercive.

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