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Lower semicontinuity of some functionals under PDE constraints: An \mathcal{A}-quasiconvex pair

Authors:

A V Demyanov

Institution:

(1) St. Petersburg State University, St.Petersburg, Russia

Abstract:

The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:

$(u,v,\chi ) \mapsto \int_\Omega {\left\{ {\chi (x) \cdot F^ + (x,u(x),v(x)) + (1 - \chi (x)) \cdot F^ - (x,u(x),v(x)) + (1 - \chi (x))} \right\}} dx,$

with respect to the convergence u_n → u in measure, v_n ⇀ v in L_p(Ω;ℝ^d) $\mathcal{A}v_n \to 0$ in W^−1,p(Ω), and χ_n ⇀ χ in L_p(Ω), where χ_n ∈ Z:= {χ ∈ L_∞(Ω): 0 ≤ χ(x) ≤ 1 for a.e. x}. Here $\mathcal{A}v = \sum\nolimits_{i = 1}^N {A^{(i)} \tfrac{{\partial v}}{{\partial x_i }}}$ is a constant-rank partial differential operator. The main result is that if the characteristic cone of $\mathcal{A}$ has the full dimension, then the l.s.c. is equivalent to the fact that the F^± are both $\mathcal{A}$ -quasiconvex and

$F^ + (x,u, \cdot ) - F^ - (x,u, \cdot ) \equiv C(x,u)$

for a.e. x ∈ Ω and for all u ∈ ℝ^d. As a corollary, we obtain several results for the functional

$(u,v,\chi ) \mapsto \int_\Omega {\chi (x) \cdot f(x,u(x),v(x))dx}$

with respect to the same convergence. We show that this functional is l.s.c. iff

$f(x,u,v) \equiv g(x,u).$

Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 100–119.

Keywords:

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