Abstract.We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including "mixed" boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.Mathematics Subject Classification. 74B20, 49K20.

Two-scale techniques are developed for sequences of mapssatisfying a linear dierential constraint Au k = 0. These, together with Γ-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type

We study weak lower semicontinuity of integral functionals in W 1;p under standard p-growth conditions, with integrands whose negative part may have p-growth as well. A characterization is obtained which, besides quasiconvexity of the integrand, involves a second condition that in general is weaker than boundary quasiconvexity at zero as defined by Ball and Marsden, although for p-homogeneous integrands, it reduces to the latter.

We present a new penalty term approximating the Ciarlet-Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet-Nečas condition. Moreover, the penalization can be chosen in such a way that for all low energy deformations, self-interpenetration is completely avoided already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.

Objective In orthopedic surgery, it is well known that the use of intrathecal morphine (ITM) leads to an improved quality of postoperative analgesia. Little is known how this improved analgesia affects the long-term course after surgery. Study design A randomized, double-blind trial. Setting Academic medical center. Subjects Forty-nine patients undergoing total hip or knee replacement surgery in spinal anesthesia. Methods Patients were randomly assigned to receive either 0.1 mg (n=16) or 0.2 mg (n=16) morphine sulfate intrathecally or physiological saline (n=17) added to 3 mL 0.5% isobaric bupivacaine for spinal anesthesia. As a function of the quality of the short-term postoperative analgesia, the effect on recovery and quality of life was evaluated at various time points up to 26 weeks after surgery. Results In both ITM groups, the additionally required postoperative systemic morphine dose was significantly reduced compared with the placebo group ( P =0.004). One week after operation, patients with ITM reported significantly less pain at rest ( P =0.01) compared to the placebo group. At discharge, in comparison with the 0.1 mg ITM and placebo group, the 0.2 mg ITM group showed a higher degree of impairment regarding pain, stiffness, and physical function of the respective joint ( P =0.02). Over the further follow-up period of 6 months after surgery, recovery and the quality of life did not differ significantly between the three study groups ( P >0.2). Conclusion Morphine (0.1 mg) as adjunct to 0.5% bupivacaine for spinal anesthesia is effective to produce a pronounced postoperative analgesia with a beneficial analgesic effect up to 1 week after surgery. With this study design, the different quality of postoperative analgesia had no effect on quality of life and recovery in patients over the 6-month follow-up period. In the medium term, ITM may induce hyperalgesic effects.

We state necessary and su cient conditions for weak lower semicontinuity of integral functionals of the form u → ∫ Ω h (x, u(x)) dx, where h is continuous and possesses a positively p-homogeneous recession function, p > , and u ∈ L p (Ω; ℝ m ) lives in the kernel of a constant-rank rst-order di erential operator A which admits an extension property. In the special case A = curl, apart from the quasiconvexity of the integrand, the recession function's quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly de ned notions of A-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in L p (Ω; ℝ m ) and approaching the kernel of A even if A does not have the extension property.

We show weak* in measures onΩ/ weak-L 1 sequential continuity of u → f (x, ∇u) : W 1,p (Ω; R m ) → L 1 (Ω), where f (x, ·) is a null Lagrangian for x ∈ Ω, it is a null Lagrangian at the boundary for x ∈ ∂Ω and |f (x, A)| ≤ C(1 + |A| p ). We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why u → det ∇u : W 1,n (Ω; R n ) → L 1 (Ω) fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant [26] need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis. This paper is inspired by the well-known example [3, Example 7.3] or [7, Example 8.6] showing that if Ω ⊂ R 2 is bounded and Lipschitz and {u k } ⊂ W 1,2 (Ω; R 2 ) weakly converges to the origin then, in general, Ω det ∇u k (x) dx → 0 which means that det ∇u k ⇀ 0 in L 1 (Ω), neither det ∇u k * ⇀ 0 in rca(Ω) (Radon measures onΩ). Contrary to that, if the sequence were bounded in W 1,p (Ω; R 2 ) for p > 2 then {det ∇u k } k∈N would weakly tend to zero in L 1 (Ω). Therefore, a natural question arises which functions f : R m×n → R, |f (A)| ≤ C(1 + |A| p ), have the property that u → f (∇u) is (weakly,weakly*) sequentially continuous as maps from W 1,p (Ω; R m ) to rca(Ω), p > 1. It is obvious that such functions must be quasiaffine, i.e., f is an affine function of all subdeterminants of its argument [7], however, as the above mentioned example shows, it is far from being sufficient. It turns out that this question is intimately related to concentrations of {|∇u k | p } k∈N ⊂ L 1 (Ω) at the boundary of Ω and that, for a general domain Ω, f must also depend on x ∈ Ω. We also show that the notion of quasiconvexity at the boundary, introduced in [4] to study necessary conditions for local minimizers of variational integral functionals plays a key role in our analysis.The plan of the paper is as follows. After introducing necessary notation we recall the notions of quasiconvexity and quasiconvexity at the boundary. Then we explicitly characterize all functions which, together with their negative multiple, are quasiconvex at the boundary. These are here called null Lagrangians at the boundary. Our characterization is a slight adaptation of the result of P. Sprenger [32] which does not seems to be well-known to the calculus-of-variations community. We state our main result Theorem 3.1 using a * This work was supported by the grants P201/12/0671 and P201/10/0357 (GAČR).

No abstract