-
Previous Article
A generalization of Douady's formula
- DCDS Home
- This Issue
-
Next Article
On the extension and smoothing of the Calabi flow on complex tori
$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
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 |
$\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$ |
$u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$ |
$ \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}$ |
$\mathcal{D}$ |
$\mathcal{D}$ |
$W^{1,∞}$ |
$\mathrm{H}$ |
$n≠ N$ |
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, 2nd 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, 2nd 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. |
[1] |
Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760 |
[2] |
Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313 |
[3] |
Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098 |
[4] |
G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 |
[5] |
Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621 |
[6] |
Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084 |
[7] |
Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025 |
[8] |
Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015 |
[9] |
Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383 |
[10] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3497-3528. doi: 10.3934/dcdss.2020442 |
[11] |
Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307 |
[12] |
Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083 |
[13] |
Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961 |
[14] |
Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473 |
[15] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks and Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003 |
[16] |
Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 |
[17] |
Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations and Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015 |
[18] |
Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118 |
[19] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763 |
[20] |
Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]