\mathcal{D}-solutions to the system of vectorial Calculus of Variations in L^∞ via the singular value problem

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December 2017, 37(12): 6165-6181. doi: 10.3934/dcds.2017266

$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem

Gisella Croce ^1, , Nikos Katzourakis ^2,†,, and Giovanni Pisante ^3,

1.	Normandie Univ, UNIHAVRE, LMAH, 76600 Le Havre, France
2.	Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK
3.	Universitá degli Studi della Campania "Luigi Vanvitelli", Scuola Politecnica e delle Scienze di Base, Dipartimento di Matematica e Fisica, Viale Lincoln, 5, 81100 Caserta, Italy

‡ Corresponding author

† N.K. has been partially financially supported by the EPSRC grant EP/N017412/1

Received December 2016 Revised July 2017 Published August 2017

For

$\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$

and

$u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$

, consider the system

$ \label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,0. \tag{1}$

We construct

$\mathcal{D}$

-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our

$\mathcal{D}$

-solutions are

$W^{1,∞}$

-submersions and are obtained without any convexity hypotheses for

$\mathrm{H}$

, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions

$n≠ N$

Keywords: Vectorial Calculus of Variations in L^∞, generalised solutions, fully nonlinear systems, ∞-Laplacian, young measures, singular value problem, Baire Category method, convex integration.

Mathematics Subject Classification: 35D99, 35F05.

Citation: Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266

References:

[1]	H. Abugirda and N. Katzourakis, Existence of 1D vectorial Absolute Minimisers in $L^∞$ under minimal assumptions, Proceedings of the AMS, 145 (2017), 2567-2575. doi: 10.1090/proc/13421.
[2]	L. Ambrosio and J. Malý, Very weak notions of differentiability, Proceedings of the Royal Society of Edinburgh A, 137 (2007), 447-455. doi: 10.1017/S0308210505001344.
[3]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$, Arkiv für Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326.
[4]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅱ, Arkiv für Mat., 6 (1966), 409-431. doi: 10.1007/BF02590964.
[5]	G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.
[6]	G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Arkiv für Mat., 7 (1968), 395-425. doi: 10.1007/BF02590989.
[7]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅲ, Arkiv für Mat., 7 (1969), 509-512. doi: 10.1007/BF02590888.
[8]	G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Manuscripta Math, 47 (1984), 133-151. doi: 10.1007/BF01174590.
[9]	G. Aronsson, Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=∞$, Manuscripta Math, 56 (1986), 135-158. doi: 10.1007/BF01172152.
[10]	G. Aronsson, M. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bulletin of the AMS, New Series, 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.
[11]	E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3.
[12]	E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^{∞}$ functionals, Arch. Rational Mech. Analysis, 157 (2001), 255-283. doi: 10.1007/PL00004239.
[13]	E. N. Barron, R. Jensen and C. Wang, Lower semicontinuity of $L^{∞}$ functionals, Ann. I. H. Poincaré AN, 18 (2001), 495-517. doi: 10.1016/S0294-1449(01)00070-1.
[14]	A. C. Barroso, G. Croce and A. Ribeiro, Sufficient conditions for existence of solutions to vectorial differential inclusions and applications, Houston J. Math., 39 (2013), 929-967.
[15]	C. Le Bris and P. L. Lions, Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications, Ann. di Mat. Pura ed Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4.
[16]	L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings, Journal of Differential Equations, 253 (2012), 851-877. doi: 10.1016/j.jde.2012.04.015.
[17]	C. Castaing, P. R. de Fitte and M. Valadier, Young Measures on Topological spaces with Applications in Control Theory and Probability Theory, Mathematics and Its Applications (Kluwer Academic Publishers, Academic Press), Academic Press, 2004. doi: 10.1007/1-4020-1964-5.
[18]	M. G. Crandall, A visit with the 1-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, 75-122, Springer Lecture notes 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3.
[19]	G. Croce, A differential inclusion: The case of an isotropic set, ESAIM Control Optim. Calc. Var., 11 (2005), 122-138. doi: 10.1051/cocv:2004035.
[20]	B. Dacorogna, Direct Methods in the Calculus of Variations, 2^nd edition, Applied Mathematical Sciences, Springer, 2008.
[21]	B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, Journal of Functional Analysis, 152 (1998), 404-446. doi: 10.1006/jfan.1997.3172.
[22]	B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1562-2.
[23]	B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case, Portugaliae Mathematica, 62 (2005), 421-436.
[24]	B. Dacorogna, G. Pisante and A. M. Ribeiro, On non quasiconvex problems of the calculus of variations, Discrete Contin. Dyn. Syst., 13 (2005), 961-983. doi: 10.3934/dcds.2005.13.961.
[25]	B. Dacorogna and A. M. Ribeiro, Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 907-921. doi: 10.1017/S0308210500003541.
[26]	B. Dacorogna and C. Tanteri, On the different convex hulls of sets involving singular values, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1261-1280. doi: 10.1017/S0308210500027311.
[27]	R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995.
[28]	L. C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, 2nd edition, 2010. doi: 10.1090/gsm/019.
[29]	L. C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC press, 1992.
[30]	L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Metric Spaces, De Gruyter, 2012. doi: 10.1515/9783110280517.
[31]	I. Fonseca and G. Leoni, Modern methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, 2007.
[32]	R. A. Horn and Ch. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.
[33]	N. Katzourakis, $L^∞$-variational problems for maps and the Aronsson PDE system, J. Differential Equations, 253 (2012), 2123-2139. doi: 10.1016/j.jde.2012.05.012.
[34]	N. Katzourakis, $∞$-minimal submanifolds, Proceedings of the AMS, 142 (2014), 2797-2811. doi: 10.1090/S0002-9939-2014-12039-9.
[35]	N. Katzourakis, On the structure of $∞$-harmonic maps, Communications in PDE, 39 (2014), 2091-2124. doi: 10.1080/03605302.2014.920351.
[36]	N. Katzourakis, Explicit $2D$ $∞$-harmonic maps whose interfaces have junctions and corners, Comptes Rendus Acad. Sci. Paris, Ser. I, 351 (2013), 677-680. doi: 10.1016/j.crma.2013.07.028.
[37]	N. Katzourakis, Optimal $∞$-quasiconformal immersions, ESAIM Control, Opt. and Calc. Var., 21 (2015), 561-582. doi: 10.1051/cocv/2014038.
[38]	N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in $L^∞$ and some linear elliptic systems, Comm. on Pure and Appl. Anal., 14 (2015), 313-327. doi: 10.3934/cpaa.2015.14.313.
[39]	N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in $L^∞$, Springer Briefs in Mathematics, 2015. doi: 10.1007/978-3-319-12829-0.
[40]	N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, Journal of Differential Equations, 263 (2017), 641-686. doi: 10.1016/j.jde.2017.02.048.
[41]	N. Katzourakis, Absolutely minimising generalised solutions to the equations of vectorial Calculus of Variations in $L^∞$, Calculus of Variations and PDE, 56 (2017), 1-25. doi: 10.1007/s00526-016-1099-z.
[42]	N. Katzourakis, A new characterisation of $∞$-Harmonic and $p$-Harmonic maps via affine variations in $L^∞$, Electronic Journal of Differential Equations, 2017 (2017), 1-19.
[43]	N. Katzourakis, Solutions of vectorial Hamilton-Jacobi equations are rank-one Absolute Minimisers in $L^∞$, Advances in Nonlinear Analysis, in press.
[44]	N. Katzourakis, Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients, preprint, arXiv: 1507.03042.
[45]	N. Katzourakis and J. Manfredi, Remarks on the validity of the maximum principle for the $∞$-Laplacian, Le Matematiche, 71 (2016), 63-74.
[46]	N. Katzourakis and T. Pryer, On the numerical approximation of $∞$-Harmonic mappings Nonlinear Differential Equations and Applications, 23 (2016), Art. 51, 23 pp. doi: 10.1007/s00030-016-0415-9.
[47]	B. Kirchheim, Rigidity and geometry of microstructures, in Issue 16 of Lecture notes, Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, 2003.
[48]	B. Kirchheim, Deformations with finitely many gradients and stability of convex hulls, Comptes Rendus de l'Académie des Sciences, Séries I, Mathematics, 332 (2001), 289-294. doi: 10.1016/S0764-4442(00)01792-4.
[49]	S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, (1996), 239-251.
[50]	S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math., 157 (2003), 715-742. doi: 10.4007/annals.2003.157.715.
[51]	P. Pedregal, Parametrized Measures and Variational Principles Birkhäuser, 1997. doi: 10.1007/978-3-0348-8886-8.
[52]	G. Pisante, Sufficient conditions for the existence of viscosity solutions for nonconvex Hamiltonians, SIAM J. Math. Anal., 36 (2004), 186-203. doi: 10.1137/S0036141003426902.
[53]	S. Sheffield and C. K. Smart, Vector-valued optimal Lipschitz extensions, Comm. Pure Appl. Math., 65 (2012), 128-154. doi: 10.1002/cpa.20391.
[54]	M. Valadier, Young measures, Methods of Nonconvex Analysis, 1446 (1990), 152-188. doi: 10.1007/BFb0084935.

show all references

References:

[1]	H. Abugirda and N. Katzourakis, Existence of 1D vectorial Absolute Minimisers in $L^∞$ under minimal assumptions, Proceedings of the AMS, 145 (2017), 2567-2575. doi: 10.1090/proc/13421.
[2]	L. Ambrosio and J. Malý, Very weak notions of differentiability, Proceedings of the Royal Society of Edinburgh A, 137 (2007), 447-455. doi: 10.1017/S0308210505001344.
[3]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$, Arkiv für Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326.
[4]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅱ, Arkiv für Mat., 6 (1966), 409-431. doi: 10.1007/BF02590964.
[5]	G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.
[6]	G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Arkiv für Mat., 7 (1968), 395-425. doi: 10.1007/BF02590989.
[7]	G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅲ, Arkiv für Mat., 7 (1969), 509-512. doi: 10.1007/BF02590888.
[8]	G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Manuscripta Math, 47 (1984), 133-151. doi: 10.1007/BF01174590.
[9]	G. Aronsson, Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=∞$, Manuscripta Math, 56 (1986), 135-158. doi: 10.1007/BF01172152.
[10]	G. Aronsson, M. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bulletin of the AMS, New Series, 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.
[11]	E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3.
[12]	E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^{∞}$ functionals, Arch. Rational Mech. Analysis, 157 (2001), 255-283. doi: 10.1007/PL00004239.
[13]	E. N. Barron, R. Jensen and C. Wang, Lower semicontinuity of $L^{∞}$ functionals, Ann. I. H. Poincaré AN, 18 (2001), 495-517. doi: 10.1016/S0294-1449(01)00070-1.
[14]	A. C. Barroso, G. Croce and A. Ribeiro, Sufficient conditions for existence of solutions to vectorial differential inclusions and applications, Houston J. Math., 39 (2013), 929-967.
[15]	C. Le Bris and P. L. Lions, Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications, Ann. di Mat. Pura ed Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4.
[16]	L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings, Journal of Differential Equations, 253 (2012), 851-877. doi: 10.1016/j.jde.2012.04.015.
[17]	C. Castaing, P. R. de Fitte and M. Valadier, Young Measures on Topological spaces with Applications in Control Theory and Probability Theory, Mathematics and Its Applications (Kluwer Academic Publishers, Academic Press), Academic Press, 2004. doi: 10.1007/1-4020-1964-5.
[18]	M. G. Crandall, A visit with the 1-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, 75-122, Springer Lecture notes 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3.
[19]	G. Croce, A differential inclusion: The case of an isotropic set, ESAIM Control Optim. Calc. Var., 11 (2005), 122-138. doi: 10.1051/cocv:2004035.
[20]	B. Dacorogna, Direct Methods in the Calculus of Variations, 2^nd edition, Applied Mathematical Sciences, Springer, 2008.
[21]	B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, Journal of Functional Analysis, 152 (1998), 404-446. doi: 10.1006/jfan.1997.3172.
[22]	B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1562-2.
[23]	B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case, Portugaliae Mathematica, 62 (2005), 421-436.
[24]	B. Dacorogna, G. Pisante and A. M. Ribeiro, On non quasiconvex problems of the calculus of variations, Discrete Contin. Dyn. Syst., 13 (2005), 961-983. doi: 10.3934/dcds.2005.13.961.
[25]	B. Dacorogna and A. M. Ribeiro, Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 907-921. doi: 10.1017/S0308210500003541.
[26]	B. Dacorogna and C. Tanteri, On the different convex hulls of sets involving singular values, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1261-1280. doi: 10.1017/S0308210500027311.
[27]	R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995.
[28]	L. C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, 2nd edition, 2010. doi: 10.1090/gsm/019.
[29]	L. C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC press, 1992.
[30]	L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Metric Spaces, De Gruyter, 2012. doi: 10.1515/9783110280517.
[31]	I. Fonseca and G. Leoni, Modern methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, 2007.
[32]	R. A. Horn and Ch. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.
[33]	N. Katzourakis, $L^∞$-variational problems for maps and the Aronsson PDE system, J. Differential Equations, 253 (2012), 2123-2139. doi: 10.1016/j.jde.2012.05.012.
[34]	N. Katzourakis, $∞$-minimal submanifolds, Proceedings of the AMS, 142 (2014), 2797-2811. doi: 10.1090/S0002-9939-2014-12039-9.
[35]	N. Katzourakis, On the structure of $∞$-harmonic maps, Communications in PDE, 39 (2014), 2091-2124. doi: 10.1080/03605302.2014.920351.
[36]	N. Katzourakis, Explicit $2D$ $∞$-harmonic maps whose interfaces have junctions and corners, Comptes Rendus Acad. Sci. Paris, Ser. I, 351 (2013), 677-680. doi: 10.1016/j.crma.2013.07.028.
[37]	N. Katzourakis, Optimal $∞$-quasiconformal immersions, ESAIM Control, Opt. and Calc. Var., 21 (2015), 561-582. doi: 10.1051/cocv/2014038.
[38]	N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in $L^∞$ and some linear elliptic systems, Comm. on Pure and Appl. Anal., 14 (2015), 313-327. doi: 10.3934/cpaa.2015.14.313.
[39]	N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in $L^∞$, Springer Briefs in Mathematics, 2015. doi: 10.1007/978-3-319-12829-0.
[40]	N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, Journal of Differential Equations, 263 (2017), 641-686. doi: 10.1016/j.jde.2017.02.048.
[41]	N. Katzourakis, Absolutely minimising generalised solutions to the equations of vectorial Calculus of Variations in $L^∞$, Calculus of Variations and PDE, 56 (2017), 1-25. doi: 10.1007/s00526-016-1099-z.
[42]	N. Katzourakis, A new characterisation of $∞$-Harmonic and $p$-Harmonic maps via affine variations in $L^∞$, Electronic Journal of Differential Equations, 2017 (2017), 1-19.
[43]	N. Katzourakis, Solutions of vectorial Hamilton-Jacobi equations are rank-one Absolute Minimisers in $L^∞$, Advances in Nonlinear Analysis, in press.
[44]	N. Katzourakis, Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients, preprint, arXiv: 1507.03042.
[45]	N. Katzourakis and J. Manfredi, Remarks on the validity of the maximum principle for the $∞$-Laplacian, Le Matematiche, 71 (2016), 63-74.
[46]	N. Katzourakis and T. Pryer, On the numerical approximation of $∞$-Harmonic mappings Nonlinear Differential Equations and Applications, 23 (2016), Art. 51, 23 pp. doi: 10.1007/s00030-016-0415-9.
[47]	B. Kirchheim, Rigidity and geometry of microstructures, in Issue 16 of Lecture notes, Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, 2003.
[48]	B. Kirchheim, Deformations with finitely many gradients and stability of convex hulls, Comptes Rendus de l'Académie des Sciences, Séries I, Mathematics, 332 (2001), 289-294. doi: 10.1016/S0764-4442(00)01792-4.
[49]	S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, (1996), 239-251.
[50]	S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math., 157 (2003), 715-742. doi: 10.4007/annals.2003.157.715.
[51]	P. Pedregal, Parametrized Measures and Variational Principles Birkhäuser, 1997. doi: 10.1007/978-3-0348-8886-8.
[52]	G. Pisante, Sufficient conditions for the existence of viscosity solutions for nonconvex Hamiltonians, SIAM J. Math. Anal., 36 (2004), 186-203. doi: 10.1137/S0036141003426902.
[53]	S. Sheffield and C. K. Smart, Vector-valued optimal Lipschitz extensions, Comm. Pure Appl. Math., 65 (2012), 128-154. doi: 10.1002/cpa.20391.
[54]	M. Valadier, Young measures, Methods of Nonconvex Analysis, 1446 (1990), 152-188. doi: 10.1007/BFb0084935.

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[18]	Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
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[20]	Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

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