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Model problems from nonlinear elasticity: partial regularity results. (English) Zbl 1221.35132

Summary: In this paper we prove that every weak and strong local minimizer \(u\in{W^{1,2}(\Omega,\mathbb{R}^3)}\) of the functional \(I(u)=\int_\Omega| Du|^2+f(\text{Adj}\,Du)+g(\det Du)\), where \(u: \Omega\subset\mathbb{R}^3\to \mathbb{R}^3\), \(f\) grows like \(|\text{Adj}\,Du|^p\), \(g\) grows like \(|\det Du|^q\) and \(1<q<p<2\), is \(C^{1,\alpha}\) on an open subset \(\Omega_0\) of \(\Omega\) such that \(\text{meas}(\Omega\setminus \Omega_0)=0\). Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case \(p=q\leq 2\) is also treated for weak local minimizers.

MSC:

35J50 Variational methods for elliptic systems
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
74B20 Nonlinear elasticity
74G40 Regularity of solutions of equilibrium problems in solid mechanics
74G65 Energy minimization in equilibrium problems in solid mechanics
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References:

[1] E. Acerbi and N. Fusco , A regularity theorem for minimizers of quasiconvex integrals . Arch. Rational Mech. Anal. 99 ( 1987 ) 261 - 281 . Zbl 0627.49007 · Zbl 0627.49007 · doi:10.1007/BF00284509
[2] E. Acerbi and N. Fusco , Regularity for minimizers of non-quadratic functionals: the case \(1&lt;p&lt;2\) . J. Math. Anal. Appl. 140 ( 1989 ) 115 - 135 . Zbl 0686.49004 · Zbl 0686.49004 · doi:10.1016/0022-247X(89)90098-X
[3] J.M. Ball , Convexity conditions and existence theorems in nonlinear elasticity . Arch. Rational Mech. Anal. 63 ( 1977 ) 337 - 403 . Zbl 0368.73040 · Zbl 0368.73040 · doi:10.1007/BF00279992
[4] J.M. Ball , Some open problem in elasticity , in Geometry, Mechanics and dynamics, Springer, New York ( 2002 ) 3 - 59 . Zbl 1054.74008 · Zbl 1054.74008 · doi:10.1007/0-387-21791-6_1
[5] M. Carozza , N. Fusco and G. Mingione , Partial regularity of minimizers of quasiconvex integrals with subquadratic growth . Annali Mat. Pura Appl. 175 ( 1998 ) 141 - 164 . Zbl 0960.49025 · Zbl 0960.49025 · doi:10.1007/BF01783679
[6] M. Carozza and A. Passarelli di Napoli , A regularity theorem for minimizers of quasiconvex integrals the case \(1&lt;p&lt;2\) . Proc. Roy. Soc. Edinburgh 126A ( 1996 ) 1181 - 1199 . Zbl 0955.49021 · Zbl 0955.49021 · doi:10.1017/S0308210500023350
[7] M. Carozza and A. Passarelli di Napoli , Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth . Proc. Roy. Soc Edinburgh 133A ( 2003 ) 1249 - 1262 . Zbl 1059.49033 · Zbl 1059.49033 · doi:10.1017/S0308210500002900
[8] B. Dacorogna , Direct methods in the calculus of variations . Appl. Math. Sci. 78, Springer Verlag ( 1989 ). MR 990890 | Zbl 0703.49001 · Zbl 0703.49001
[9] L.C. Evans , Quasiconvexity and partial regularity in the calculus of variations . Arch. Rational Mech. Anal. 95 ( 1986 ) 227 - 252 . Zbl 0627.49006 · Zbl 0627.49006 · doi:10.1007/BF00251360
[10] N. Fusco and J. Hutchinson , Partial regularity in problems motivated by nonlinear elasticity . SIAM J. Math. 22 ( 1991 ) 1516 - 1551 . Zbl 0744.35014 · Zbl 0744.35014 · doi:10.1137/0522098
[11] N. Fusco and J. Hutchinson , Partial regularity and everywhere continuity for a model problem from nonlinear elasticity . J. Australian Math. Soc. 57 ( 1994 ) 149 - 157 . Zbl 0864.35032 · Zbl 0864.35032
[12] M. Giaquinta , Multiple integrals in the calculus of variations and nonlinear elliptic systems . Ann. Math. Stud. 105 Princeton Univ. Press ( 1983 ). MR 717034 | Zbl 0516.49003 · Zbl 0516.49003
[13] M. Giaquinta and G. Modica , Partial regularity of minimizers of quasiconvex integrals . Ann. Inst. H. Poincaré, Analyse non linéaire 3 ( 1986 ) 185 - 208 . Numdam | Zbl 0594.49004 · Zbl 0594.49004
[14] E. Giusti , Metodi diretti in calcolo delle variazioni . U.M.I. ( 1994 ). · Zbl 0942.49002
[15] J. Kristensen and A. Taheri , Partial regularity of strong local minimizers in the multidimensional calculus of variations . Arch. Rational Mech. Anal. 170 ( 2003 ) 63 - 89 . Zbl 1030.49040 · Zbl 1030.49040 · doi:10.1007/s00205-003-0275-4
[16] A. Passarelli di Napoli , A regularity result for a class of polyconvex functionals . Ricerche di Matematica XLVIII ( 1999 ) 379 - 393 . Zbl 0947.35052 · Zbl 0947.35052
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