1,545 research outputs found
Liftings, Young measures, and lower semicontinuity
This work introduces liftings and their associated Young measures as new
tools to study the asymptotic behaviour of sequences of pairs for
under weak* convergence. These
tools are then used to prove an integral representation theorem for the
relaxation of the functional
to the space . Lower semicontinuity results of this type were first obtained
by Fonseca and M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later
improved by a number of authors, but our theorem is valid under more natural,
essentially optimal, hypotheses than those currently present in the literature,
requiring principally that be Carath\'eodory and quasiconvex in the final
variable. The key idea is that liftings provide the right way of localising
in the and variables simultaneously under weak*
convergence. As a consequence, we are able to implement an optimal
measure-theoretic blow-up procedure.Comment: 75 pages. Updated to correct a series of minor typos/ inaccuracies.
The statement and proof of Theorem have also been amended- subsequent steps
relying upon the Theorem did not require updatin
Piecewise affine approximations for functions of bounded variation
BV functions cannot be approximated well by piecewise constant functions, but
this work will show that a good approximation is still possible with
(countably) piecewise affine functions. In particular, this approximation is
area-strictly close to the original function and the -difference
between the traces of the original and approximating functions on a substantial
part of the mesh can be made arbitrarily small. Necessarily, the mesh needs to
be adapted to the singularities of the BV function to be approximated, and
consequently, the proof is based on a blow-up argument together with explicit
constructions of the mesh. In the case of -Sobolev functions
we establish an optimal -error estimate for approximation by
piecewise affine functions on uniform regular triangulations. The piecewise
affine functions are standard quasi-interpolants obtained by mollification and
Lagrange interpolation on the nodes of triangulations, and the main new
contribution here compared to for instance Cl\'{e}ment (RAIRO Analyse
Num\'{e}rique 9 (1975), no.~R-2, 77--84) and Verf\"{u}rth (M2AN
Math.~Model.~Numer.~Anal. 33 (1999), no. 4, 695-713) is that our error
estimates are in the -norm rather than merely the
-norm.Comment: 14 pages, 1 figur
Relaxation for partially coercive integral functionals with linear growth
We prove an integral representation theorem for the -relaxation of the functional , where () is a bounded Lipschitz domain, to the space under very general assumptions: we require principally that is Carathéodory, that the partial coercivity and linear growth bound , hold, where is a continuous function satisfying a weak monotonicity condition, and that is quasi-convex in the final variable. Our result is the first that applies to integrands which are unbounded in the -variable and, therefore, allows for the treatment of many problems from applications. Such functionals are out of reach of the classical blowup approach introduced by Fonseca and Müller [Arch. Ration. Mech. Anal., 123 (1993), pp. 1--49]. Our proof relies on an intricate truncation construction (in the - and -arguments simultaneously) made possible by the theory of liftings developed in a previous paper by the authors [Arch. Ration. Mech. Anal., 232 (2019), pp. 1227--1328], and features techniques which could be of use for other problems involving -dependent integrands
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