0584247 20250317090823.320240315235959.9 85187669803 001222132200001 10.1016/j.na.2024.113523 29 s. P cav_un_epca*02573310362-546X243Elsevier Hadamard inequality Quasiconvexity at the boundary cav_un_auth*0465031 Bevan J. GB 33 K cav_un_auth*0101142 Kružík Martin UTIA-B Matematická teorie rozhodování Department of Decision Making Theory MTR MTR Department of Decision Making Theory 34 Ústav teorie informace a automatizace AV ČR, v. v. i. cav_un_auth*0292941 Valdman Jan UTIA-B Matematická teorie rozhodování Department of Decision Making Theory MTR MTR Department of Decision Making Theory 33 Ústav teorie informace a automatizace AV ČR, v. v. i. http://library.utia.cas.cz/separaty/2024/MTR/kruzik-0584247.pdf https://www.sciencedirect.com/science/article/pii/S0362546X24000427?via%3Dihub IEES R3 193278 Royal Society International Exchange GB cav_un_auth*0465032 Let 𝑄 be a Lipschitz domain in R𝑛 and let 𝑓 ∈ 𝐿∞(𝑄). We investigate conditions under which the functional 𝐼𝑛(𝜑) = ∫𝑄 |∇𝜑|𝑛 + 𝑓(𝑥)det ∇𝜑 d𝑥 obeys 𝐼𝑛 ≥ 0 for all 𝜑 ∈ 𝑊1,𝑛 0 (𝑄,R𝑛), an inequality that we refer to as Hadamard-in-the-mean, or (HIM). We prove that there are piecewise constant 𝑓 such that (HIM) holds and is strictly stronger than the best possible inequality that can be derived using the Hadamard inequality 𝑛 𝑛2 |det 𝐴| ≤ |𝐴|𝑛 alone. When 𝑓 takes just two values, we find that (HIM) holds if and only if the variation of 𝑓 in 𝑄 is at most 2𝑛 𝑛2. For more general 𝑓, we show that (i) it is both the geometry of the ‘jump sets’ as well as the sizes of the ‘jumps’ that determine whether (HIM) holds and (ii) the variation of 𝑓 can be made to exceed 2𝑛 𝑛2, provided 𝑓 is suitably chosen. Specifically, in the planar case 𝑛 = 2 we divide 𝑄 into three regions {𝑓 = 0} and {𝑓 = ±𝑐}, and prove that as long as {𝑓 = 0} ‘insulates’ {𝑓 = 𝑐} from {𝑓 = −𝑐} sufficiently, there is 𝑐 > 2 such that (HIM) holds. Perhaps surprisingly, (HIM) can hold even when the insulation region {𝑓 = 0} enables the sets {𝑓 = ±𝑐} to meet in a point. As part of our analysis, and in the spirit of the work of Mielke and Sprenger (1998), we give new examples of functions that are quasiconvex at the boundary. WOS BA 10000 10100 10102 2025 3 RVO:67985556 https://hdl.handle.net/11104/0353335 S 113523 A MATHEMATICS|MATHEMATICS.APPLIED 1.5 0.3 14.9 0.00951 0.97 11493 141 1.532 31.11 1.53 895 Q1 1.300 161 Q1 74.6 0.95 Q1 85.7 2024 85187669803 SCOPUS PUBMED 001222132200001 WOS cav_un_epca*0257331 Nonlinear Analysis: Theory, Methods & Applications Roč. 243 č. 1 2024 0362-546X 1873-5215 Elsevier