Abstract. Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function, we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

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BackgroundTo investigate long-term refractive outcomes, wavefront aberrations and quality of life after small incision lenticule extraction (SMILE) for moderate to high myopia.MethodsA total of 26 patients (47 eyes) with preoperative mean spherical equivalent (SE) of -6.30 ± 1.47 diopters (D) who underwent SMILE were recruited. The measurements included uncorrected distance visual acuity (UDVA), corrected distance visual acuity (CDVA), manifest refraction, wavefront aberrations, and quality of life.ResultsAt 4 years postoperatively, UDVA was better than or equal to 20/20 in 92 % of eyes. The efficacy index was 1.07 ± 0.16. 89 % of eyes were within ± 0.5 D of the intended refractive target. No eye lost any Snellen lines. The safety index was 1.16 ± 0.14. No significant changes of SE occurred among postoperative follow-ups at months 1, 3, 6 and years 1, 2, 4 (P > 0.05, Scheffe test). Higher-order aberrations, coma, spherical aberration and higher-order astigmatism increased postoperatively, and no significant changes of aberrations were detected among the 1-month, 6-month or 4-year follow-ups postoperatively (37 eyes). Compared to the spectacles group, the surgery group showed a significantly higher total score on quality of life (45.71 ± 2.61 vs 39.96 ± 3.56, P < 0.001).ConclusionsSMILE provides a predictable and stable correction of moderate to high myopia as documented by long-term follow-up.

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Dual extremum principles are established in this paper for the variational boundary-value problem of elasto-perfect pl/tsticity with large deformation. There exists a duality gap between the primal and dual variational problems. Our application to nonlinear limit analysis yields a pair of dual bounding theorems for the safety factor, when the gap has the right sign. It is proved that both the upper and lower bounds directly depend on the properties of the dual gap function.AMS subject classifications ~1980L 73Exx. 73Gxx.

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