1,720,985 research outputs found
Semicontinuity and supremal representation in the Calculus of Variations.
No full textWe study the weak* lower semicontinuity properties of functionals of the form
F(u)=\supess_{x \in \Og} f(x,Du (x))
where \Og is a bounded open set of and
Without a continuity assumption on we show that the {\sl supremal} functional is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in general it is not possible to represent through the level convex envelope of
Relaxation and gamma-convergence of supremal functionals
No full textWe prove that the Gamma-limit in L^1_μ of a sequence of supremal functionals
of the form F_k(u) = μ-ess sup
f_k(x, u) is itself a supremal functional. We
show by a counterexample that, in general, the function which represents the Gamma-
lim F(·,B) of a sequence of functionals F_k(u,B) = μ-ess sup_B f_k(x, u) can depend
on the set B and we give a necessary and sufficient condition to represent F in
the supremal form F(u,B) = μ-ess sup_B f(x, u). As a corollary, if f represents a
supremal functional, then the level convex envelope of f represents its weak* lower
semicontinuous envelope
"Supremal Representation of L^{infty} Functionals"
No full textWe study the weak* lower semicontinuity properties of functionals of the form
F(u)=\supess_{x \in \Og} f(x,Du (x))
where \Og is a bounded open set of and
Without a continuity assumption on we show that the {\sl supremal} functional is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in general it is not possible to represent through the level convex envelope of
Standing waves for a class of nonlinear Schr"odinger equations with potentials in .
No full textWe prove the existence of standing waves
to the following family of nonlinear Sch\"odinger equations:
provided that is small,
when ,
when and
is assumed to have a sublevel with positive and finite measure
On a Minimization Problem Involving the Critical Sobolev Exponent.
No full textWe study the following minimization problem:
in any dimension and under suitable assumptions on .
\noindent Mainly we assume that
belongs to the Lorentz space
and the set
has positive Lebesgue measure. Notice that this last condition is
satisfied when the set has a nontrivial interior part
(in fact this is the typical assumption imposed in the literature on
the set )
On the lower semicontinuity and approximation of functionals
No full textIn this paper we show that if the supremal functional
F(V,B) = ess sup x∈B
f(x, DV (x))
is sequentially weak* lower semicontinuous on W1,∞(B, Rd) for every open set B ⊆ Ω (where Ω is a fixed open set of RN ), then f(x, ·) is rank-one level convex for a.e x ∈ Ω. Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the Lp-approximation of a supremal functional F via Γ-convergence when f is a non-negative and coercive Carath ́eodory function
Semicontinuity and relaxation of -functionals
No full textFixed a bounded open set \Og of , we completely characterize the weak* lower semicontinuity of functionals of the form
F(u,A)=\supess_{x \in A} f(x,u(x),Du (x))
defined for every and for every open subset A\subset \Om.
Without a continuity assumption on we show that the {\sl supremal} functional is weakly* lower semicontinuous if and only if it can be represented through a {\sl level convex} function. Then we study the properties of the lower semicontinuous envelope of . A complete relaxation theorem is shown in the case where is a continuous function. In the case is only a Carath\'eodory function,
we show that coincides with the level convex envelope of
The role of intrinsic distances in the relaxation of L∞-functionals
No full textWe consider a supremal functional of the form
F(u) = ess sup
x∈Ω
f(x,Du(x))
where Ω ⊆ RN is a regular bounded open set, u ∈ W1,∞(Ω) and f is a Borel
function. Assuming that the intrinsic distances dλ
F (x, y) := sup
{
u(x) − u(y) :
F(u) ≤ λ
}
are locally equivalent to the euclidean one for every λ > infW1,∞(Ω) F,
we give a description of the sublevel sets of the weak∗-lower semicontinuous
envelope of F in terms of the sub-level sets of the difference quotient functionals
RdλF(u) := supx̸=yu(x)−u(y)dλF (x,y) . As a consequence we prove that the relaxed
functional of positive 1-homogeneous supremal functionals coincides with Rd1F.
Moreover, for a more general supremal functional F (a priori non coercive), we
prove that the sublevel sets of its relaxed functionals with respect to the weak∗
topology, the weak∗ convergence and the uniform convergence are convex. The
proof of these results relies both on a deep analysis of the intrinsic distances
associated to F and on a careful use of variational tools such as Γ-convergence
On the lower semicontinuity of supremal functional under differential constraint
Full text linkWe study the weak* lower semicontinuity of supremal functionals
under a differential constraint that is described by a constant-rank partial differential operator A. The notion of A-Young quasiconvexity, which is introduced here, provides a sufficient condition when the supremand function is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity.
Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity
The class of functionals which can be represented by a supremum.
No full textWe give a characterization of all lower semicontinuous functionals on L^\infty_\mu which can be represented in
the form \mu - sup \{ f(x; u) : x \in A\}. We also show by a counterexample that the representation above may
fail if the lower semicontinuity condition is dropped
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