Abstract
We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-known results, which help prove the well-posedness of the various sets of equations we study throughout this book.
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- 1.
In a vector space, a scalar product (⋅, ⋅) exhibits the following properties:
-
It is linear with respect to the first variable:
\(\forall a_1,a_2\in \mathbb {C},\ \forall v_1,v_2,w\in V,\ (a_1v_1+a_2v_2,w) = a_1(v_1,w) + a_2(v_2,w)\).
-
It is antilinear with respect to the second variable:
\(\forall a_1,a_2\in \mathbb {C},\ \forall v,w_1,w_2\in V,\ (v,a_1w_1+a_2w_2) = \overline {a_1}(v,w_1) + \overline {a_2}(v,w_2)\).
-
It is Hermitian:
\(\forall v,w\in V,\ (v,w) = \overline {(w,v)}\).
-
It is positive-definite:
∀v ∈ V ∖{0}, (v, v) > 0.
Then, \(\|\cdot \|\ :\ V\rightarrow \mathbb {R}\), defined by ∥v∥ = (v, v)1∕2, is a norm on V . Furthermore, the Cauchy-Schwarz inequality holds: ∀v, w ∈ V, |(v, w)|≤∥v∥ ∥w∥.
-
- 2.
More generally, one may define the resolvent and spectrum of an unbounded operator A from D(A) ⊂ X to X. In this case, the resolvent is
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \rho(A)=\{\lambda\in\mathbb{C}\ :\ (A-\lambda I_X)(D(A))\mbox{ is dense in }X\,;\\ &\displaystyle &\displaystyle \hskip 15mm (A-\lambda I_X)^{-1} \mbox{ exists and is continuous from } (A-\lambda I_X)(D(A))\mbox{ to } X \}\,; \end{array} \end{aligned} $$the spectrum is \(\sigma (A)=\mathbb {C}\setminus \rho (A)\), and it can further be decomposed into the disjoint union of the point spectrum , the continuous spectrum and the residual spectrum (see [93, Chapter VIII, §1] for details). As a rule, the notions of a continuous or residual spectrum will not be needed for the study of operators in this book.
- 3.
Some of these results are consequences of the Fredholm alternative, which we choose to state hereafter within the framework of Hilbert spaces.
- 4.
V ′ can also be called the antidual space. We choose the denomination dual space, which also applies for vector spaces defined over \(\mathbb {R}\), and continuous linear forms. Given v ∈ V , f v : w↦(v, w) V defines an element of V ′. According to the Riesz Theorem 4.2.1 below, v↦f v is a bijective isometry from V to V ′. In addition, V ′ can be made into a Hilbert space by defining its scalar product via \((f_v,f_w)_{V'}=(v,w)_V\), for all v, w ∈ V .
- 5.
A Hilbert basis of V is a countable set \((e_k)_{k\in \mathbb {N}}\) of elements of V , such that, for all k, ℓ, (e k , e ℓ ) V = δ kℓ , and
is dense in V . Then, for all v ∈ V , one has \(v=\sum _{k\in \mathbb {N}}(v,e_k)_V\, e_k\) and \(\|v\|{ }_V^2=\sum _{k\in \mathbb {N}}(v,e_k)_V^2\) (Bessel-Parseval identity). - 6.
Let us explain briefly this construction when the imbedding W ⊂ c V is compact; this condition will hold in all the cases encountered in this book. Using Corollary 4.5.12 below, which is a straightforward consequence of Theorem 4.1.20, one constructs a Hilbert basis \((e_k)_{k\in \mathbb {N}}\) of V whose elements belong to W, and a nondecreasing sequence of strictly positive numbers \((\mu _k)_{k\in \mathbb {N}}\) tending to + ∞ such that:
$$\displaystyle \begin{aligned} \forall w\in W,\quad (e_k, w)_W = \mu_k^2\, (e_k, w)_V \,. \end{aligned}$$Clearly, ∥e k ∥ W = μ k , thus the space W can be alternatively defined as
$$\displaystyle \begin{aligned} W = \{ w = \sum_{k\in\mathbb{N}} w_k\, e_k \in V : \sum_{k\in\mathbb{N}} \mu_k^2\, |w_k|{}^2 < +\infty \} = D(\varLambda), \quad \text{where:}\quad \varLambda = \sum_{k\in\mathbb{N}} \mu_k\, P_k, \end{aligned}$$and P k denotes the projection onto \( \operatorname {{\mathrm {span}}}\{e_k\}\). Then, for any \(\alpha \in \mathbb {R}^+\), one defines the unbounded operator power Λ α as
$$\displaystyle \begin{aligned} D(\varLambda^\alpha) = \{ w = \sum_{k\in\mathbb{N}} w_k\, e_k \in V : \sum_{k\in\mathbb{N}} \mu_k^{2\alpha}\, |w_k|{}^2 < +\infty \} \quad \text{and:}\quad \varLambda^\alpha = \sum_{k\in\mathbb{N}} \mu_k^\alpha\, P_k. \end{aligned}$$When the imbedding W ⊂ V is not compact, the above discrete sums are replaced with Stieltjes integrals that take into account the whole spectrum (see [207, §XI]).
- 7.
In the compact imbedding framework, the next two propositions follow immediately from Definition 4.1.21 and footnote6.
- 8.
A sesquilinear form is linear with respect to the first variable, and antilinear with respect to the second variable.
- 9.
When the sesquilinear form in Problem (4.3) is Hermitian (V = W), if the associated operator is Fredholm, then its index is always equal to 0.
- 10.
For practical applications, it allows one to consider initial data that do not verify the constraints that the solution fulfills afterwards.
- 11.
It is equivalently written as
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} \displaystyle \forall \varphi\in \mathcal{D}(]0,T[),\ \forall\mathtt{v}\in\mathcal{V},\\ \displaystyle \int_0^T\left\{(\mathtt{u}(t),\mathtt{v})_{\mathcal{H}}\,\varphi''(t) + a(\mathtt{u}(t),\mathtt{v})\,\varphi(t)\right\}dt = \int_0^T(\mathtt{f}(t),\mathtt{v})_{\mathcal{H}}\,\varphi(t)\,dt. \end{array} \right. \end{aligned}$$ - 12.
Unless g(t) is appropriately regular. More precisely, see (4.27-top) below: g(t) should be regular enough, so that the second term on the right-hand side can be included in the first term by suitably modifying f(t).
- 13.
Cf. the discussion in footnote 6, p. 153.
- 14.
Since g can be any element of Q′, one has to assume that B is surjective. If g = 0, then this inf-sup condition could be dropped to formulate the Helmholtz-like problem set in K. However, it is useful in Proposition 4.5.8.
- 15.
Because the operator A is an isomorphism, one has λ≠0, as it holds that Au = λ u in V ′, with u≠0.
- 16.
One can check that \(T_H^*=i_{V\rightarrow H}\circ (A^{-1})^\dagger \circ i_{H\rightarrow V'}\).
is dense in V . Then, for all v ∈ V , one has References
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Assous, F., Ciarlet, P., Labrunie, S. (2018). Abstract Mathematical Framework. In: Mathematical Foundations of Computational Electromagnetism. Applied Mathematical Sciences, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-70842-3_4
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