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Semicontinuity Problems in the Calculus of Variations
 

Summary: Semicontinuity Problems in the Calculus
of Variations
EMILIO ACERBI & NICOLA Fusco
Communicated by J. SERRIN
Introduction
Let f: R a -+ R be a twice continuously differentiable function, and for
u E Wl'l(a, b) set
b
F(u; a, b) ----f f(x, u(x), u'(x)) dx.
a
In a paper of TONELLX[17] it is proved that the functional Fis lower semieontinuous
(lsc) in the topology of L~(a, b) if and only if the function f is convex in the last
variable. Later, several authors generalized this result: among the many theorems
obtained, in which x is allowed to belong to R" and considerably less regularity
on fis required, we recall particularly Theorem 12 of SERRIN [15], in which for
the first time differentiability conditions on f are dropped, and the following
result due to MARCELLINI & SBORDONE [11]:
If f: R" · · -+ R satisfies:
(i) f is measurable in x, and continuous in (s, ~), and
(ii) 0 <~f(x, s, ~) <=g(x, Is], [~]),

  

Source: Acerbi, Emilio - Dipartimento di Matematica, Universitą degli Studi di Parma
Fusco, Nicola - Dipartimento di Matematica e Applicazioni, Universitą degli Studi di Napoli "Federico II"

 

Collections: Mathematics