Advanced Search
Article Contents
Article Contents

On the validity of the Euler-Lagrange system

Abstract Related Papers Cited by
  • 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 $s>1$.
    Mathematics Subject Classification: Primary: 49N15, 49N60; Secondary: 49N99.


    \begin{equation} \\ \end{equation}
  • [1]

    E. Acerbi and N. Fusco, Partial regularity under anisotropic $(p, q)$ growth conditions, J. Diff. Eq., 107 (1994), 46-67.doi: 10.1006/jdeq.1994.1002.


    J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), 129-147.


    J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Ration. Mech. Anal., 90 (1985), 325-388.doi: 10.1007/BF00276295.


    M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Mathematics, 1818. Springer-Verlag, Berlin, 2003.doi: 10.1007/b12308.


    G. Bonfanti, A. Cellina and M. Mazzola, The higher integrability and the validity of the Euler-Lagrange equation for solutions to variational problems, SIAM J. Control Optim., 50 (2012), 888-899.doi: 10.1137/110820890.


    G. Bonfanti and A. Cellina, The nonoccurrence of the Lavrentiev phenomenon for a class of variational functionals, SIAM J. Control Optim., 51 (2013), 1639-1650.doi: 10.1137/12086618X.


    M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals, Ann. Inst. Henri Poincaré, Anal. Non Linaire, 28 (2011), 395-411.doi: 10.1016/j.anihpc.2011.02.005.


    M. Carozza, J. Kristensen and A. Passarelli di Napoli, Regularity of minimizers of autonomous convex variational integrals, Ann. Sc. Norm. Super. Pisa Cl. Sci. (V), to appear.


    M. Carozza, G. Moscariello and A. Passarelli di Napoli, Regularity results via duality for minimizers of degenerate functionals, Asympt. Anal., 44 (2005), 221-235.


    M. Carozza and A. Passarelli di Napoli, Regularity for minimizers of degenerate elliptic functionals, J. Nonlinear Convex Anal., 7 (2006), 375-383.


    I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28, SIAM, Philadelphia, 1999.doi: 10.1137/1.9781611971088.


    L. Esposito, F. Leonetti and G. Mingione, Sharp higher integrability for minimizers of integral functionals with $(p, q)$ growth, J. Differential Equations, 204 (2004), 5-55.doi: 10.1016/j.jde.2003.11.007.


    L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p,q)$ growth, Forum Mathematicum, 14 (2002), 245-272.doi: 10.1515/form.2002.011.


    I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 309-338.doi: 10.1016/S0294-1449(97)80139-4.


    M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245-248.doi: 10.1007/BF01158049.


    E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.doi: 10.1142/9789812795557.


    T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161.doi: 10.1515/crll.1994.454.143.


    G. Kresin and V. Maz'ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Mathematical Surveys and Monographs 183, American Mathematical Society, Providence, RI, 2012.


    J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in $W^{1,1}$ and $\BV$, Arch. Ration. Mech. Anal., 197 (2010), 539-598. Erratum: ibid 203 (2012), 693-700.


    J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal., 180 (2006), 331-398.doi: 10.1007/s00205-005-0402-5.


    J. Kristensen and G. Mingione, Boundary regularity in variational problems, Arch. Ration. Mech. Anal., 198 (2010), 369-455.doi: 10.1007/s00205-010-0294-x.


    J. L. Lewis, On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations, 18 (1993), 1515-1537.doi: 10.1080/03605309308820984.


    P. Marcellini, Un example de solution discontinue d'un probéme variationel dans le cas scalaire, Preprint Ist. U. Dini, Firenze, 1987-88.


    P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284.doi: 10.1007/BF00251503.


    P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Cl. Sci., 23 (1996), 1-25.


    P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Diff. Eq., 221 (2006), 412-443.doi: 10.1016/j.jde.2004.11.011.


    G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426.doi: 10.1007/s10778-006-0110-3.


    A. Passarelli di Napoli and F. Siepe, A regularity result for a class of anisotropic systems, Rend. Ist. Mat di Trieste, (1997), 13-31.


    J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Sup. Cl. Sci., 18 (1964), 385-387.


    V. Šverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Nat. Acad. Sci. USA, 99 (2002), 15269-15276.doi: 10.1073/pnas.222494699.


    V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116.


    W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Maths. 120, Springer-Verlag, 1989.doi: 10.1007/978-1-4612-1015-3.

  • 加载中

Article Metrics

HTML views() PDF downloads(216) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint