By L.P. Lebedev, I. I. Vorovich
This ebook covers sensible research and its purposes to continuum mechanics. The mathematical fabric is taken care of in a non-abstract demeanour and is absolutely illuminated via the underlying mechanical rules. The presentation is concise yet whole, and is meant for experts in continuum mechanics who desire to comprehend the mathematical underpinnings of the self-discipline. Graduate scholars and researchers in arithmetic, physics, and engineering will locate this ebook important. workouts and examples are integrated all through with distinctive options supplied within the appendix.
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Extra info for Functional analysis in mechanics
If w = 0 then (w, w) = 0. If (w, w) = 0 then, on Ω, ∂2w = 0, ∂x2 ∂2w = 0, ∂x∂y ∂2w = 0. 10 Some Energy Spaces 41 It follows that w(x, y) = a1 + a2 x + a3 y where the ai are constants. 17) then, w(x, y) = 0. Hence P1 is satisﬁed. 15), and it is evident that P3 is also satisﬁed. 14) is an inner product space; its completion in the corresponding metric is the energy space EP C for a clamped plate. Let us consider some properties of the elements of EP C . It was shown that m0 Ω 2 ∂2w ∂x2 +2 ≤ ∂2w ∂x∂y 2 2 2 ∂2w ∂y 2 + dx dy Dαβγδ ργδ (w)ραβ (w) dx dy ≡ (w, w).
The reader should take a moment to write down the Schwarz inequality in both cases, and to write down the inner product for L2 (Ω) over the reals. Most importantly, the energy spaces we introduced earlier are all inner product spaces. 2) is a metric space. We called this an energy space for the clamped bar. 3) 0 l 1/2 2 B(x)[y (x)] dx 0 on this space. We have d(y, z) = y − z . 10 Some Energy Spaces 33 C (2) (0, l); the reader should construct an example). To have a complete space, we must apply the completion theorem.
Another method is connected with the Fourier transformation of functions. In the case of W m,2 (Ω), it is necessary to extend functions of C (m) (Ω) outside Ω in such a way that they belong to C m (Rn ) and W m,2 (Rn ). Then using the Fourier transformation u ˆ(y) = (2π)−n/2 Rn e−ix·y u(x) dx1 · · · dxn along with the facts that u(x) L2 (Rn ) = u ˆ(y) L2 (Rn ) and Dα u(x) = (iy1 )α1 · · · (iyn )αn u ˆ(y) for u ∈ L2 (Rn ), we can present the norm in W m,2 (Ω) in the form u(x) 2 W m,2 (Rn ) y1α1 · · · ynαn u ˆ(y) = 2 L2 (Rn ) .