The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

37The following theorems are standard results of functional analysis and will be usedto prove solvability of boundary integral equations studied in the next sections. In thetheorems we assume that X denotes a Hilbert space.Theorem 3.18 (Lax-Milgram Lemma). Let A: X → X ∗ denote a linear, bounded andX-elliptic operator. Then for any f ∈ X ∗ there exists a unique element u ∈ X satisfyingMoreover, there holds the estimatewith the ellipticity constant c A 1 .Au = f.∥u∥ X ≤ 1c A ∥f∥ X ∗1Proof. For the proof of the Lax-Milgram Lemma see [18], proof following Theorem 3.4 orany standard functional analysis textbook.Theorem 3.19 (Fredholm Alternative). Let K : X → X denote a compact operator. Eitherthe homogeneous equation(I − K)u = 0has a nontrivial solution u ∈ X or the inhomogeneous problem(I − K)u = fhas a unique solution u ∈ X for all f ∈ X. In the latter case there exists a constant c ∈ R +such that∥u∥ X ≤ c∥f∥ X for all f ∈ X.Proof. For the proof of the Fredholm Alternative see, e.g., [7], Theorem 2.2.9.Theorem 3.20. Let A: X → X ∗ denote a linear, bounded and coercive operator and letA be injective, i.e., from Au = 0 it follows u = 0. Then for every f ∈ X ∗ there exists aunique solution to the equationAu = f. (3.19)Moreover, there exists a constant c ∈ R + such that∥u∥ X ≤ c∥f∥ X ∗ for all f ∈ X ∗ . (3.20)Proof. From coercivity of A we know that there exists a compact operator C such that thelinear operator D := A + C : X → X ∗ is X-elliptic. From the Lax-Milgram Lemma 3.18we get that there exists the inverse operator D −1 : X ∗ → X. Therefore, we obtain thatthe equation (3.19) is equivalent toBu = D −1 f