December  2021, 20(12): 4271-4305. doi: 10.3934/cpaa.2021160

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

* Corresponding author

Received  February 2021 Revised  April 2021 Published  December 2021 Early access  September 2021

In this paper we prove a partial Hölder regularity result for weak solutions
$ u:\Omega\to \mathbb{R}^N $
,
$ N\geq 2 $
, to non-autonomous elliptic systems with general growth of the type:
$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $
The crucial point is that the operator
$ a $
satisfies very weak regularity properties and a general growth, while the inhomogeneity
$ b $
has a controllable growth.
Citation: Teresa Isernia, Chiara Leone, Anna Verde. Partial regularity result for non-autonomous elliptic systems with general growth. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4271-4305. doi: 10.3934/cpaa.2021160
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.  Google Scholar

[2]

V. BögeleinF. DuzaarJ. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. DieningD. LengelerB. 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.  Google Scholar

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L. DieningB. Stroffolini and A. Verde, Everywhere regularity of functionals with $\phi$-growth, Manuscripta Math., 129 (2009), 449-481.  doi: 10.1007/s00229-009-0277-0.  Google Scholar

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L. DieningB. 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.  Google Scholar

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P. Di GironimoL. 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.  Google Scholar

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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.  Google Scholar

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F. DuzaarJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[20]

T. IserniaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

C. B. Morrey, Quasi–convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[2]

V. BögeleinF. DuzaarJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. DieningD. LengelerB. 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.  Google Scholar

[9]

L. DieningB. Stroffolini and A. Verde, Everywhere regularity of functionals with $\phi$-growth, Manuscripta Math., 129 (2009), 449-481.  doi: 10.1007/s00229-009-0277-0.  Google Scholar

[10]

L. DieningB. 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.  Google Scholar

[11]

P. Di GironimoL. 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.  Google Scholar

[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.  Google Scholar

[13]

F. DuzaarJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

T. IserniaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

C. B. Morrey, Quasi–convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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