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Periodic solutions for a class of second-order differential delay equations
Partial regularity result for non-autonomous elliptic systems with general growth
1. | Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, via brecce bianche 12, 60131 Ancona, Italy |
2. | Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico II, via Cinthia, 80126 Napoli, Italy |
$ u:\Omega\to \mathbb{R}^N $ |
$ N\geq 2 $ |
$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $ |
$ a $ |
$ b $ |
References:
[1] |
V. Bögelein,
Partial regularity for minimizers of discontinuous quasi–convex integrals with degeneracy, J. Differ. Equ., 252 (2012), 1052-1100.
doi: 10.1016/j.jde.2011.09.031. |
[2] |
V. Bögelein, F. Duzaar, J. Habermann and C. Scheven,
Partial Hölder continuity for discontinuous elliptic problems with VMO–coefficients, Proc. Lond. Math. Soc., 103 (2011), 371-404.
doi: 10.1112/plms/pdr009. |
[3] |
D. Breit and A. Verde,
Quasiconvex variational functionals in Orlicz–Sobolev spaces, Ann. Mat. Pura Appl., 192 (2013), 255-271.
doi: 10.1007/s10231-011-0222-1. |
[4] |
P. Celada and J. Ok, Partial regularity for non–autonomous degenerate quasi–convex functionals with general growth, Nonlinear Anal., 194 (2020), 111473, 36 pp.
doi: 10.1016/j.na.2019.02.026. |
[5] |
E. De Giorgi, Frontiere orientate di misura minima Seminario di Matematica della, in Scuola Normale Superiore di Pisa, Editrice Tecnico Scienti ca, Pisa, 1961. |
[6] |
L. Diening and F. Ettwein,
Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20 (2008), 523-556.
doi: 10.1515/FORUM.2008.027. |
[7] |
L. Diening and C. Kreuzer,
Linear convergence of an adaptive finite element method for the $p$-Laplacian equation, SIAM J. Num. Anal., 46 (2008), 614-638.
doi: 10.1137/070681508. |
[8] |
L. Diening, D. Lengeler, B. Stroffolini and A. Verde,
Partial regularity for minimizer of quasi-convex functional with general growth, SIAM J. Math. Anal., 44 (2012), 3594-3616.
doi: 10.1137/120870554. |
[9] |
L. Diening, B. Stroffolini and A. Verde,
Everywhere regularity of functionals with $\phi$-growth, Manuscripta Math., 129 (2009), 449-481.
doi: 10.1007/s00229-009-0277-0. |
[10] |
L. Diening, B. Stroffolini and A. Verde,
The $\varphi$-harmonic approximation and the regularity of $\varphi$-harmonic maps, J. Differ. Equ., 253 (2012), 1943-1958.
doi: 10.1016/j.jde.2012.06.010. |
[11] |
P. Di Gironimo, L. Esposito and L. Sgambati,
A remark on $L^{2, \lambda}$ regularity for minimizers of quadratic functionals, Manuscripta Math., 113 (2004), 143-151.
doi: 10.1007/s00229-003-0429-6. |
[12] |
F. Duzaar and J. F. Grotowski,
Optimal interior partial regularity for nonlinear elliptic systems: The method of $\mathcal{A} $-harmonic approximation, Manuscripta Math., 103 (2000), 267-298.
doi: 10.1007/s002290070007. |
[13] |
F. Duzaar, J. F. Grotowski and M. Kronz,
Regularity of almost minimizers of quasi–convex variational integrals with subquadratic growth, Ann. Mat. Pura Appl., 184 (2005), 421-448.
doi: 10.1007/s10231-004-0117-5. |
[14] |
F. Duzaar and M. Kronz,
Regularity of $\omega$-minimizers of quasi-convex variational integrals with polynomial growth, Differ. Geom. Appl., 17 (2002), 139-152.
doi: 10.1016/S0926-2245(02)00104-3. |
[15] |
F. Duzaar and G. Mingione,
The $p$-harmonic approximation and the regularity of $p$-harmonic maps, Calc. Var. Partial Differ. Equ., 20 (2004), 235-256.
doi: 10.1007/s00526-003-0233-x. |
[16] |
F. Duzaar and G. Mingione,
Harmonic type approximation lemmas, J. Math. Anal. Appl., 352 (2009), 301-335.
doi: 10.1016/j.jmaa.2008.09.076. |
[17] |
M. Foss and G. Mingione,
Partial continuity for elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 471-503.
doi: 10.1016/j.anihpc.2007.02.003. |
[18] |
E. Giusti and M. Miranda,
Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Ration. Mech. Anal., 31 (1968/69), 173-184.
doi: 10.1007/BF00282679. |
[19] |
C. Goodrich, G. Scilla and B. Stroffolini, Partial Hölder continuity for minimizers of discontinuous quasiconvex integrals with VMO coefficients and general growth, preprint, 2021. |
[20] |
T. Isernia, C. Leone and A. Verde,
Partial regularity results for asymptotic quasiconvex functionals with general growth, Ann. Acad. Sci. Fenn. Math., 41 (2016), 817-844.
doi: 10.5186/aasfm.2016.4155. |
[21] |
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. |
[22] |
J. Kristensen and G. Mingione,
The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal., 184 (2007), 341-369.
doi: 10.1007/s00205-006-0036-2. |
[23] |
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. |
[24] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate parabolic equations, Nonlinear Anal., 14 (1990), 501-524.
doi: 10.1016/0362-546X(90)90038-I. |
[25] |
P. Marcellini,
Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.
doi: 10.1007/BF00251503. |
[26] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with $p, q$–growth conditions, J. Differ. Equ., 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[27] |
P. Marcellini,
Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 1-25.
|
[28] |
C. B. Morrey,
Quasi–convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.
|
[29] |
J. Ok,
Partial Hölder regularity for elliptic systems with non-standard growth, J. Funct. Anal., 274 (2018), 723-768.
doi: 10.1016/j.jfa.2017.11.014. |
[30] |
M. Ragusa and A. Tachikawa,
Partial regularity of the minimizers of quadratic functionals with VMO coefficients, J. Lond. Math. Soc., 72 (2005), 609-620.
doi: 10.1112/S002461070500699X. |
[31] |
M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146 Marcel Dekker, Inc., New York, 1991. |
[32] |
L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Birkhäuser Verlag, Basel, 1996.
doi: 10.1007/978-3-0348-9193-6. |
[33] |
B. Stroffolini,
Partial regularity results for quasimonotone elliptic systems with general growth, Z. Anal. Anwend., 39 (2020), 315-347.
doi: 10.4171/zaa/1662. |
[34] |
S. Z. Zheng,
Partial regularity for quasi-linear elliptic systems with VMO coefficients under the natural growth, Chinese Ann. Math. Ser. A, 29 (2008), 49-58.
|
show all references
References:
[1] |
V. Bögelein,
Partial regularity for minimizers of discontinuous quasi–convex integrals with degeneracy, J. Differ. Equ., 252 (2012), 1052-1100.
doi: 10.1016/j.jde.2011.09.031. |
[2] |
V. Bögelein, F. Duzaar, J. Habermann and C. Scheven,
Partial Hölder continuity for discontinuous elliptic problems with VMO–coefficients, Proc. Lond. Math. Soc., 103 (2011), 371-404.
doi: 10.1112/plms/pdr009. |
[3] |
D. Breit and A. Verde,
Quasiconvex variational functionals in Orlicz–Sobolev spaces, Ann. Mat. Pura Appl., 192 (2013), 255-271.
doi: 10.1007/s10231-011-0222-1. |
[4] |
P. Celada and J. Ok, Partial regularity for non–autonomous degenerate quasi–convex functionals with general growth, Nonlinear Anal., 194 (2020), 111473, 36 pp.
doi: 10.1016/j.na.2019.02.026. |
[5] |
E. De Giorgi, Frontiere orientate di misura minima Seminario di Matematica della, in Scuola Normale Superiore di Pisa, Editrice Tecnico Scienti ca, Pisa, 1961. |
[6] |
L. Diening and F. Ettwein,
Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20 (2008), 523-556.
doi: 10.1515/FORUM.2008.027. |
[7] |
L. Diening and C. Kreuzer,
Linear convergence of an adaptive finite element method for the $p$-Laplacian equation, SIAM J. Num. Anal., 46 (2008), 614-638.
doi: 10.1137/070681508. |
[8] |
L. Diening, D. Lengeler, B. Stroffolini and A. Verde,
Partial regularity for minimizer of quasi-convex functional with general growth, SIAM J. Math. Anal., 44 (2012), 3594-3616.
doi: 10.1137/120870554. |
[9] |
L. Diening, B. Stroffolini and A. Verde,
Everywhere regularity of functionals with $\phi$-growth, Manuscripta Math., 129 (2009), 449-481.
doi: 10.1007/s00229-009-0277-0. |
[10] |
L. Diening, B. Stroffolini and A. Verde,
The $\varphi$-harmonic approximation and the regularity of $\varphi$-harmonic maps, J. Differ. Equ., 253 (2012), 1943-1958.
doi: 10.1016/j.jde.2012.06.010. |
[11] |
P. Di Gironimo, L. Esposito and L. Sgambati,
A remark on $L^{2, \lambda}$ regularity for minimizers of quadratic functionals, Manuscripta Math., 113 (2004), 143-151.
doi: 10.1007/s00229-003-0429-6. |
[12] |
F. Duzaar and J. F. Grotowski,
Optimal interior partial regularity for nonlinear elliptic systems: The method of $\mathcal{A} $-harmonic approximation, Manuscripta Math., 103 (2000), 267-298.
doi: 10.1007/s002290070007. |
[13] |
F. Duzaar, J. F. Grotowski and M. Kronz,
Regularity of almost minimizers of quasi–convex variational integrals with subquadratic growth, Ann. Mat. Pura Appl., 184 (2005), 421-448.
doi: 10.1007/s10231-004-0117-5. |
[14] |
F. Duzaar and M. Kronz,
Regularity of $\omega$-minimizers of quasi-convex variational integrals with polynomial growth, Differ. Geom. Appl., 17 (2002), 139-152.
doi: 10.1016/S0926-2245(02)00104-3. |
[15] |
F. Duzaar and G. Mingione,
The $p$-harmonic approximation and the regularity of $p$-harmonic maps, Calc. Var. Partial Differ. Equ., 20 (2004), 235-256.
doi: 10.1007/s00526-003-0233-x. |
[16] |
F. Duzaar and G. Mingione,
Harmonic type approximation lemmas, J. Math. Anal. Appl., 352 (2009), 301-335.
doi: 10.1016/j.jmaa.2008.09.076. |
[17] |
M. Foss and G. Mingione,
Partial continuity for elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 471-503.
doi: 10.1016/j.anihpc.2007.02.003. |
[18] |
E. Giusti and M. Miranda,
Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Ration. Mech. Anal., 31 (1968/69), 173-184.
doi: 10.1007/BF00282679. |
[19] |
C. Goodrich, G. Scilla and B. Stroffolini, Partial Hölder continuity for minimizers of discontinuous quasiconvex integrals with VMO coefficients and general growth, preprint, 2021. |
[20] |
T. Isernia, C. Leone and A. Verde,
Partial regularity results for asymptotic quasiconvex functionals with general growth, Ann. Acad. Sci. Fenn. Math., 41 (2016), 817-844.
doi: 10.5186/aasfm.2016.4155. |
[21] |
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. |
[22] |
J. Kristensen and G. Mingione,
The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal., 184 (2007), 341-369.
doi: 10.1007/s00205-006-0036-2. |
[23] |
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. |
[24] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate parabolic equations, Nonlinear Anal., 14 (1990), 501-524.
doi: 10.1016/0362-546X(90)90038-I. |
[25] |
P. Marcellini,
Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.
doi: 10.1007/BF00251503. |
[26] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with $p, q$–growth conditions, J. Differ. Equ., 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[27] |
P. Marcellini,
Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 1-25.
|
[28] |
C. B. Morrey,
Quasi–convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.
|
[29] |
J. Ok,
Partial Hölder regularity for elliptic systems with non-standard growth, J. Funct. Anal., 274 (2018), 723-768.
doi: 10.1016/j.jfa.2017.11.014. |
[30] |
M. Ragusa and A. Tachikawa,
Partial regularity of the minimizers of quadratic functionals with VMO coefficients, J. Lond. Math. Soc., 72 (2005), 609-620.
doi: 10.1112/S002461070500699X. |
[31] |
M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146 Marcel Dekker, Inc., New York, 1991. |
[32] |
L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Birkhäuser Verlag, Basel, 1996.
doi: 10.1007/978-3-0348-9193-6. |
[33] |
B. Stroffolini,
Partial regularity results for quasimonotone elliptic systems with general growth, Z. Anal. Anwend., 39 (2020), 315-347.
doi: 10.4171/zaa/1662. |
[34] |
S. Z. Zheng,
Partial regularity for quasi-linear elliptic systems with VMO coefficients under the natural growth, Chinese Ann. Math. Ser. A, 29 (2008), 49-58.
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