Luc TARTAR Compensated Compactness with more ... - ICMS

With H-measures, it is implied by a more natural conjecture: if
u n , v n converge in L ∞ weak ⋆ and correspond to a H-measure
µ, then (*) implies that a.e. (x, t)
µ is supported at two opposite points in (ξ, τ).
There seems to be a geometrical idea behind the preceding
calculations, that some 2-forms vanish on K.
In the PDE courses which I taught at CMU, I pointed out to
a pedagogical mistake made by Laurent SCHWARTZ: when he
said that a locally integrable function f defines a distribution,
he called that distribution f, and he should have called it f dx
for emphasizing the role of dx, although in the end of his book
he mentions that there is no natural volume form on a manifold,
and he describes the currents of DE RHAM (1903–1990).
I heard Laurent SCHWARTZ make fun of one of my teachers
in physics at École Polytechnique, Louis LEPRINCE-RINGUET
(1901–2000), who had said that the Hilbert spaces used by
physicists are different from those used by mathematicians.
Laurent SCHWARTZ should have pointed out that for a complex
Hilbert space mathematicians use an Hermitian product (a, b)
which is linear in a and anti-linear in b, while physicists use
the notation of DIRAC 〈c|d〉, which is linear in d and anti-linear
in c, and explain the notation | d〉〈c | for an operator, which
mathematicians denote with a tensor product notation d ⊗ c.
DIRAC called 〈c| a “bra” and |d〉 a “ket”, and for H = L 2 a
function f ∈ L 2 is denoted |f〉, while 〈f| is an element of the
dual H ′ , namely f dx.
For V = H 1 0 (Ω) ⊂ H = L 2 (Ω), the canonical isometry of V
onto V ′ is u ↦→ −∆ u, that of H onto H ′ is u ↦→ u dx.
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