×

A regularity theorem for minimizers of quasiconvex integrals. (English) Zbl 0627.49007

Let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded set. Let \(f: {\mathbb{R}}^{nN}\to {\mathbb{R}}\) be a function of class \(C^ 2\) such that \[ | f(\xi)| \leq L(1+| \xi |^ p), \]
\[ \int_{\Omega}f(\xi +D\phi)dx\geq \int_{\Omega}[f(\xi)+\gamma (| D\phi |^ 2+| D\phi |^ p)]dx \] for all \(\xi \in {\mathbb{R}}^{nN}\) and all \(\phi \in C^ 1_ 0(\Omega;{\mathbb{R}}^ N)\) (p\(\geq 2\), L, \(\gamma =const>0)\). Then the first main result of the paper under review is as follows. Let \(u\in W^{1,p}(\Omega;{\mathbb{R}}^ N)\) satisfy \[ \int_{\Omega}f(Du)dx\leq \int_{\Omega}f(D(u+\phi))dx,\quad \phi \in W_ 0^{1,p}(\Omega;{\mathbb{R}}^ N). \] Then there exists an open set \(\Omega_ 0\subset \Omega\) such that \(meas(\Omega \setminus \Omega_ 0)=0\) and \(u\in C^{1,\mu}(\Omega_ 0;{\mathbb{R}}^ N)\) for all \(0<\mu <1\). The proof is based on the blow-up method and an approximation lemma combined with a higher integrability result for minima of certain non-coercive functionals. The above result is extended to the case when f also depends on (x,u).
Reviewer: J.Naumann

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
26B25 Convexity of real functions of several variables, generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Acerbi, E., &amp; N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145. · Zbl 0565.49010
[2] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403. · Zbl 0368.73040
[3] Eisen, G., A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals, Manuscripta Math. 27 (1979), 73–79. · Zbl 0404.28004
[4] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474. · Zbl 0441.49011
[5] Evans, L. C., Quasiconvexity and partial regularity in the calculus of variations. · Zbl 0627.49006
[6] Evans, L. C., &amp; R. F. Gariepy, Blow-up, compactness and partial regularity in the calculus of variations. · Zbl 0648.49007
[7] Fusco, N., &a · Zbl 0587.49005
[8] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, 1983. · Zbl 0516.49003
[9] Giaquinta, M., &amp; G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986), 185–208. · Zbl 0594.49004
[10] Hong, M.-C., Existence and partial regularity in the calculus of variations,
[11] Marcellini, P., Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), 1–28. · Zbl 0573.49010
[12] Meyers, N. G., Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc. 119 (1965), 125–149. · Zbl 0166.38501
[13] Acerbi, E., &amp; N. Fusco, An a
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.