Partial regularity result for non-autonomous elliptic systems with general growth

Teresa Isernia; Chiara Leone; Anna Verde

doi:10.3934/cpaa.2021160

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

Partial regularity result for non-autonomous elliptic systems with general growth

Teresa Isernia ^1,, , Chiara Leone ^2, and Anna Verde ^2,

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.

Keywords: Elliptic systems, general growth, partial regularity.

Mathematics Subject Classification: Primary: 35J47, 35B65; Secondary: 46E30.

Citation: Teresa Isernia, Chiara Leone, Anna Verde. Partial regularity result for non-autonomous elliptic systems with general growth. Communications on Pure and 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.
[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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