Abstract
The main purpose in the previous chapters has been to introduce part of the basic and classical theory of some important equations of mathematical physics. The emphasis on phenomenological aspects and the connection with a probabilistic point of view should have conveyed to the reader some intuition and feeling about the interpretation and the limits of those models.
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Notes
- 1.
See e.g. Rudin [34].
- 2.
The set of the functions which are Lebesgue integrable in A is denoted with L1(A); we will be more precise in Section 7.2.1. The set of all the functions which are Lebesgue integrable in every interval (a, b) is denoted by \( {L}_{loc}^1\left(\mathbb{R}\right) \).
- 3.
We may write(math insert)
$$ {\displaystyle {\int}_0^{+\infty}\frac{\left| \sin x\right|}{x}} dx={\displaystyle \sum_{k=1}^{\infty }{\displaystyle {\int}_{\left(k-1\right)\pi}^{k\pi}\frac{\left| \sin x\right|}{x} dx\ge {\displaystyle \sum_{k=1}^{\infty}\frac{1}{ k\pi}{\displaystyle {\int}_{\left(k-1\right)\pi}^{k\pi}\left| \sin x\right| dx={\displaystyle \sum_{k=1}^{\infty}\frac{2}{ k\pi}=+\infty .}}}}} $$ - 4.
See also Section 7.2.2 and Example 7.4.
- 5.
See e.g. Yoshida [37].
- 6.
δjk is the Kronecker symbol
- 7.
Notation: if L is linear, when no confusion arises, we may write Lx instead of L(x)
- 8.
The inner products in V and H may be different V
- 9.
That is \( a\left(x,\alpha y+\beta z\right)=\overline{\alpha}a\left(x,y\right)+\overline{\beta}a\left(x,y\right). \)
- 10.
From
$$ \left|{\displaystyle {\int}_{\varOmega}\left(u-\upsilon \right)\left({\varphi}_k-\psi \right)d\mathrm{x}}\right|\le \left|\right|u-\upsilon \left|{\Big|}_0\right|\left|{\varphi}_k-\psi \right|{\Big|}_0. $$ - 11.
For instance, let φ ∈ 𝒟 (Ω. We have, by Hőlder’s inequality:
$$ \left|{\displaystyle {\int}_{\varOmega}\left({u}_k-u\right)\varphi d\mathrm{x}}\right|\le \left|\right|{u}_k-u\left|{\Big|}_{L^p\left(\varOmega \right)}\right|\left|\varphi \right|{\Big|}_{L^q\left(\varOmega \right)} $$where q = p/ (p −1). Then, if ||u k − u|| p (Ω L → 0, also ∫Ω (u k − u) φ d X → 0, showing the convergence of {u k in 𝒟′ (Ω)
- 12.
Only a finite number of integers k belongs to the support of φ.
- 13.
Recall that φ = 0 and ∇φ = 0 on ∂B R .
- 14.
- 15.
Also H 1,2(Ω) or W 1,2 (Ω) are used.
- 16.
Rigorously: every equivalence class in H 1 (a, b) has a continuous representative in [a, b.
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Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P. (2013). Elements of Functional Analysis. In: A Primer on PDEs. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2862-3_7
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