Boundary effects and weak${}^{☆}$ lower semicontinuity for signed integral functionals on $BV$

Benešová, Barbora; Krömer, Stefan; Kružík, Martin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 513-534.

We characterize lower semicontinuity of integral functionals with respect to weak${}^{☆}$ convergence in $BV$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition – quasi-sublinear growth from below at points of the boundary – which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by Kristensen and Rindler [J. Kristensen and F. Rindler, Arch. Rat. Mech. Anal. 197 (2010) 539–598; J. Kristensen and F. Rindler, Calc. Var. 37 (2010) 29–62].

E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125–145. | DOI | Zbl

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Clarendon Press, Oxford, 2000. | Zbl

M. Baía, M. Chermisi, J. Matias and P.M. Santos, Lower semicontinuity and relaxation of signed functionals with linear growth in the context of $𝒜$-quasiconvexity. Calc. Var. Partial Differ. Equ. 47 (2013) 465–498. | DOI | Zbl

J.M. Ball and J.E. Marsden, Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86 (1984) 251–277. | DOI | Zbl

L. Beck and T. Schmidt, On the Dirichlet problem for variational integrals in $BV$. J. Reine Angew. Math. 674 (2013) 113–194. | Zbl

I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$. SIAM J. Math. Anal. 23 (1992) 1081–1098. | DOI | Zbl

I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in $BV(\Omega ,R^N)$ for integrands $f(x,u,▽u)$. Arch. Ration. Mech. Anal. 123 (1993) 1–49. | DOI | Zbl

I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. | DOI | Zbl

A. Kałamajska and M. Kružík, Oscillations and concentrations in sequences of gradients. ESAIM: COCV 14 (2008) 71–104. | Numdam | Zbl

A. Kałamajska, S. Krömer and M. Kružík, Sequential weak continuity of null lagrangians at the boundary. Calc. Var. Partial Differ. Equ. 49 (2014) 1263–1278. | DOI | Zbl

J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and BV. Arch. Ration. Mech. Anal. 197 (2010) 539–598. | DOI | Zbl

J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mat-report 1994-34, Math. Institute, Technical University of Denmark, 1994.

J. Kristensen and F. Rindler, Relaxation of signed integral functionals in BV. Calc. Var. Partial Differ. Equ. 37 (2010) 29–62. | DOI | Zbl

S. Krömer and M. Kružík, Oscillations and concentrations in sequences of gradients up to the boundary. J. Convex Anal. 20 (2013) 723–752. | Zbl

Stefan Krömer, On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals. Adv. Calc. Var. 3 (2010) 387–408. | Zbl

M. Kružík, Quasiconvexity at the boundary and concentration effects generated by gradients. ESAIM: COCV 19 (2013) 679–700. | Numdam | Zbl

A. Mielke and P. Sprenger, Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition. J. Elasticity 51 (1998) 23–41. | DOI | Zbl

C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25–53. | DOI | Zbl

F. Rindler and G. Shaw, Strictly continuous extensions and convex lower semicontinuity of functionals with linear growth. Preprint arXiv:1312.4554v2 [math.AP] (2013).

P. Sprenger, Quasikonvexität am Rande und Null-Lagrange-Funktionen in der nichtkonvexen Variationsrechnung. Ph.D. thesis, Universität Hannover (1996). | Zbl