# American Institute of Mathematical Sciences

January  2015, 14(1): 51-62. doi: 10.3934/cpaa.2015.14.51

## On the validity of the Euler-Lagrange system

 1 Università del Sannio, Piazza Arechi II - 82100 Benevento 2 Mathematical Institute, 24–29 St Giles’, University of Oxford, OX1 3LB Oxford 3 Dipartimento di Matematica e Appl. “R. Caccioppoli”, Via Cintia- Monte S.Angelo, 80126 Napoli

Received  February 2014 Revised  March 2014 Published  September 2014

The minimizers of convex integral functionals of the form \begin{eqnarray} \mathfrak{F} (v, \Omega) = \int_{\Omega} F (Dv (x)) dx, \end{eqnarray} defined on Sobolev mappings $v$ in $W^{1,1}_{g}(\Omega R^N)$ are characterized as the energy solutions to the Euler--Lagrange system for $\mathfrak{F}$. We assume that the integrands $F: R^{N\times n} \to R$ are $C^1$, convex and super--linear at infinity, and the boundary datum $g \in W^{1,1}(\Omega, R^N)$ must satisfy $F(sDg) \in L^1(\Omega )$ for some number $s>1$.
Citation: Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
##### References:
 [1] E. Acerbi and N. Fusco, Partial regularity under anisotropic $(p, q)$ growth conditions,, \emph{J. Diff. Eq.}, 107 (1994), 46. doi: 10.1006/jdeq.1994.1002. Google Scholar [2] J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, \emph{J. Convex Anal.}, 4 (1997), 129. Google Scholar [3] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation,, \emph{Arch. Ration. Mech. Anal.}, 90 (1985), 325. doi: 10.1007/BF00276295. Google Scholar [4] M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions,, Lecture Notes in Mathematics, (1818). doi: 10.1007/b12308. Google Scholar [5] G. Bonfanti, A. Cellina and M. Mazzola, The higher integrability and the validity of the Euler-Lagrange equation for solutions to variational problems,, \emph{SIAM J. Control Optim.}, 50 (2012), 888. doi: 10.1137/110820890. Google Scholar [6] G. Bonfanti and A. Cellina, The nonoccurrence of the Lavrentiev phenomenon for a class of variational functionals,, \emph{SIAM J. Control Optim.}, 51 (2013), 1639. doi: 10.1137/12086618X. Google Scholar [7] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals,, \emph{Ann. Inst. Henri Poincar\'e, 28 (2011), 395. doi: 10.1016/j.anihpc.2011.02.005. Google Scholar [8] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Regularity of minimizers of autonomous convex variational integrals,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci. (V)}, (). Google Scholar [9] M. Carozza, G. Moscariello and A. Passarelli di Napoli, Regularity results via duality for minimizers of degenerate functionals,, \emph{Asympt. Anal.}, 44 (2005), 221. Google Scholar [10] M. Carozza and A. Passarelli di Napoli, Regularity for minimizers of degenerate elliptic functionals,, \emph{J. Nonlinear Convex Anal.}, 7 (2006), 375. Google Scholar [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Classics in Applied Mathematics 28, (1999). doi: 10.1137/1.9781611971088. Google Scholar [12] L. Esposito, F. Leonetti and G. Mingione, Sharp higher integrability for minimizers of integral functionals with $(p, q)$ growth,, \emph{J. Differential Equations}, 204 (2004), 5. doi: 10.1016/j.jde.2003.11.007. Google Scholar [13] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p,q)$ growth,, \emph{Forum Mathematicum}, 14 (2002), 245. doi: 10.1515/form.2002.011. Google Scholar [14] I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 14 (1997), 309. doi: 10.1016/S0294-1449(97)80139-4. Google Scholar [15] M. Giaquinta, Growth conditions and regularity, a counterexample,, \emph{Manuscripta Math.}, 59 (1987), 245. doi: 10.1007/BF01158049. Google Scholar [16] E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003). doi: 10.1142/9789812795557. Google Scholar [17] T. Iwaniec and C. Sbordone, Weak minima of variational integrals,, \emph{J. Reine Angew. Math.}, 454 (1994), 143. doi: 10.1515/crll.1994.454.143. Google Scholar [18] G. Kresin and V. Maz'ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems,, Mathematical Surveys and Monographs \textbf{183}, 183 (2012). Google Scholar [19] J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and $\BV$,, \emph{Arch. Ration. Mech. Anal.}, 197 (2010), 539. Google Scholar [20] J. Kristensen and G. Mingione, The singular set of minima of integral functionals,, \emph{Arch. Ration. Mech. Anal.}, 180 (2006), 331. doi: 10.1007/s00205-005-0402-5. Google Scholar [21] J. Kristensen and G. Mingione, Boundary regularity in variational problems,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 369. doi: 10.1007/s00205-010-0294-x. Google Scholar [22] J. L. Lewis, On very weak solutions of certain elliptic systems,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1515. doi: 10.1080/03605309308820984. Google Scholar [23] P. Marcellini, Un example de solution discontinue d'un probéme variationel dans le cas scalaire,, Preprint Ist. U. Dini, (): 1987. Google Scholar [24] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions,, \emph{Arch. Ration. Mech. Anal.}, 105 (1989), 267. doi: 10.1007/BF00251503. Google Scholar [25] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions,, \emph{Ann. Scuola Norm. Sup. Cl. Sci., 23 (1996), 1. Google Scholar [26] P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth,, \emph{J. Diff. Eq.}, 221 (2006), 412. doi: 10.1016/j.jde.2004.11.011. Google Scholar [27] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations,, \emph{Appl. Math.}, 51 (2006), 355. doi: 10.1007/s10778-006-0110-3. Google Scholar [28] A. Passarelli di Napoli and F. Siepe, A regularity result for a class of anisotropic systems,, \emph{Rend. Ist. Mat di Trieste}, (1997), 13. Google Scholar [29] J. Serrin, Pathological solutions of elliptic differential equations,, \emph{Ann. Sc. Norm. Sup. Cl. Sci., 18 (1964), 385. Google Scholar [30] V. Šverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals,, \emph{Proc. Nat. Acad. Sci. USA}, 99 (2002), 15269. doi: 10.1073/pnas.222494699. Google Scholar [31] V. V. Zhikov, On some variational problems,, \emph{Russian J. Math. Phys.}, 5 (1997), 105. Google Scholar [32] W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Maths. 120, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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##### References:
 [1] E. Acerbi and N. Fusco, Partial regularity under anisotropic $(p, q)$ growth conditions,, \emph{J. Diff. Eq.}, 107 (1994), 46. doi: 10.1006/jdeq.1994.1002. Google Scholar [2] J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures,, \emph{J. Convex Anal.}, 4 (1997), 129. Google Scholar [3] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation,, \emph{Arch. Ration. Mech. Anal.}, 90 (1985), 325. doi: 10.1007/BF00276295. Google Scholar [4] M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions,, Lecture Notes in Mathematics, (1818). doi: 10.1007/b12308. Google Scholar [5] G. Bonfanti, A. Cellina and M. Mazzola, The higher integrability and the validity of the Euler-Lagrange equation for solutions to variational problems,, \emph{SIAM J. Control Optim.}, 50 (2012), 888. doi: 10.1137/110820890. Google Scholar [6] G. Bonfanti and A. Cellina, The nonoccurrence of the Lavrentiev phenomenon for a class of variational functionals,, \emph{SIAM J. Control Optim.}, 51 (2013), 1639. doi: 10.1137/12086618X. Google Scholar [7] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals,, \emph{Ann. Inst. Henri Poincar\'e, 28 (2011), 395. doi: 10.1016/j.anihpc.2011.02.005. Google Scholar [8] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Regularity of minimizers of autonomous convex variational integrals,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci. (V)}, (). Google Scholar [9] M. Carozza, G. Moscariello and A. Passarelli di Napoli, Regularity results via duality for minimizers of degenerate functionals,, \emph{Asympt. Anal.}, 44 (2005), 221. Google Scholar [10] M. Carozza and A. Passarelli di Napoli, Regularity for minimizers of degenerate elliptic functionals,, \emph{J. Nonlinear Convex Anal.}, 7 (2006), 375. Google Scholar [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Classics in Applied Mathematics 28, (1999). doi: 10.1137/1.9781611971088. Google Scholar [12] L. Esposito, F. Leonetti and G. Mingione, Sharp higher integrability for minimizers of integral functionals with $(p, q)$ growth,, \emph{J. Differential Equations}, 204 (2004), 5. doi: 10.1016/j.jde.2003.11.007. Google Scholar [13] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p,q)$ growth,, \emph{Forum Mathematicum}, 14 (2002), 245. doi: 10.1515/form.2002.011. Google Scholar [14] I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 14 (1997), 309. doi: 10.1016/S0294-1449(97)80139-4. Google Scholar [15] M. Giaquinta, Growth conditions and regularity, a counterexample,, \emph{Manuscripta Math.}, 59 (1987), 245. doi: 10.1007/BF01158049. Google Scholar [16] E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003). doi: 10.1142/9789812795557. Google Scholar [17] T. Iwaniec and C. Sbordone, Weak minima of variational integrals,, \emph{J. Reine Angew. Math.}, 454 (1994), 143. doi: 10.1515/crll.1994.454.143. Google Scholar [18] G. Kresin and V. Maz'ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems,, Mathematical Surveys and Monographs \textbf{183}, 183 (2012). Google Scholar [19] J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and $\BV$,, \emph{Arch. Ration. Mech. Anal.}, 197 (2010), 539. Google Scholar [20] J. Kristensen and G. Mingione, The singular set of minima of integral functionals,, \emph{Arch. Ration. Mech. Anal.}, 180 (2006), 331. doi: 10.1007/s00205-005-0402-5. Google Scholar [21] J. Kristensen and G. Mingione, Boundary regularity in variational problems,, \emph{Arch. Ration. Mech. Anal.}, 198 (2010), 369. doi: 10.1007/s00205-010-0294-x. Google Scholar [22] J. L. Lewis, On very weak solutions of certain elliptic systems,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1515. doi: 10.1080/03605309308820984. Google Scholar [23] P. Marcellini, Un example de solution discontinue d'un probéme variationel dans le cas scalaire,, Preprint Ist. U. Dini, (): 1987. Google Scholar [24] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions,, \emph{Arch. Ration. Mech. Anal.}, 105 (1989), 267. doi: 10.1007/BF00251503. Google Scholar [25] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions,, \emph{Ann. Scuola Norm. Sup. Cl. Sci., 23 (1996), 1. Google Scholar [26] P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth,, \emph{J. Diff. Eq.}, 221 (2006), 412. doi: 10.1016/j.jde.2004.11.011. Google Scholar [27] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations,, \emph{Appl. Math.}, 51 (2006), 355. doi: 10.1007/s10778-006-0110-3. Google Scholar [28] A. Passarelli di Napoli and F. Siepe, A regularity result for a class of anisotropic systems,, \emph{Rend. Ist. Mat di Trieste}, (1997), 13. Google Scholar [29] J. Serrin, Pathological solutions of elliptic differential equations,, \emph{Ann. Sc. Norm. Sup. Cl. Sci., 18 (1964), 385. Google Scholar [30] V. Šverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals,, \emph{Proc. Nat. Acad. Sci. USA}, 99 (2002), 15269. doi: 10.1073/pnas.222494699. Google Scholar [31] V. V. Zhikov, On some variational problems,, \emph{Russian J. Math. Phys.}, 5 (1997), 105. Google Scholar [32] W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Maths. 120, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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