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August  2019, 39(8): 4863-4873. doi: 10.3934/dcds.2019198

Regularity and weak comparison principles for double phase quasilinear elliptic equations

Giuseppe Riey

Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Rende (CS), Italy

Received  November 2018 Revised  February 2019 Published  May 2019

We consider the Euler equation of functionals involving a term of the form
$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $
with
$ 1<p<q<p+1 $
 and
$ a(x)\geq 0 $
. We prove weak comparison principle and summability results for the second derivatives of solutions.
Keywords: Quasilinear elliptic equations, p-Laplace operator, regularity, comparison principle, double phase functionals.
Mathematics Subject Classification: Primary: 35B05, 35B65, 35J92; Secondary: 35D10.
Citation: Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198
References:
[1]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z.  Google Scholar

[2]

D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. Google Scholar

[3]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496.   Google Scholar

[4]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.  Google Scholar

[5]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.  Google Scholar

[6]

G. Cupini, F. Leonetti and E. Mascolo, Existence of weak solutions for elliptic systems with $p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658.  doi: 10.5186/aasfm.2015.4035.  Google Scholar

[7]

G. Cupini, P. Marcellini and E. Mascolo, Existence for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724.   Google Scholar

[8]

L. Damascelli, Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[9]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar

[10]

L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159.  doi: 10.1007/s00526-005-0337-6.  Google Scholar

[11]

E. Di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[12]

L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55.  doi: 10.1016/j.jde.2003.11.007.  Google Scholar

[13]

A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.  Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[15]

T. Leonori, A. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.  doi: 10.1007/s10231-016-0600-9.  Google Scholar

[16]

P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503.  Google Scholar

[17]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.  Google Scholar

[18]

C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, SIAM J. Math. Anal., 48 (2016), 2059-2075.  doi: 10.1137/16M1058546.  Google Scholar

[19]

N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056. doi: 10.1073/pnas.51.6.1055.  Google Scholar

[20]

G. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.   Google Scholar

[21]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122.  doi: 10.1007/BF02413623.  Google Scholar

[22]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[23]

G. Riey, Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9.   Google Scholar

[24]

G. Riey and B. Sciunzi, A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858.  doi: 10.1051/cocv/2017040.  Google Scholar

[25]

B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334.  doi: 10.1007/s00030-007-5047-7.  Google Scholar

[26]

B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp. doi: 10.1142/S0219199714500138.  Google Scholar

[27]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[28]

N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.   Google Scholar

[29]

V. V. Zhykov, Averaging of functional of the calculus of variations and elasticity theory, Izk. Akad. Nauk. SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar

[30]

V. V. Zhykov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

show all references

References:
[1]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z.  Google Scholar

[2]

D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. Google Scholar

[3]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496.   Google Scholar

[4]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.  Google Scholar

[5]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.  Google Scholar

[6]

G. Cupini, F. Leonetti and E. Mascolo, Existence of weak solutions for elliptic systems with $p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658.  doi: 10.5186/aasfm.2015.4035.  Google Scholar

[7]

G. Cupini, P. Marcellini and E. Mascolo, Existence for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724.   Google Scholar

[8]

L. Damascelli, Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[9]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar

[10]

L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159.  doi: 10.1007/s00526-005-0337-6.  Google Scholar

[11]

E. Di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[12]

L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55.  doi: 10.1016/j.jde.2003.11.007.  Google Scholar

[13]

A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.  Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[15]

T. Leonori, A. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.  doi: 10.1007/s10231-016-0600-9.  Google Scholar

[16]

P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503.  Google Scholar

[17]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.  Google Scholar

[18]

C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, SIAM J. Math. Anal., 48 (2016), 2059-2075.  doi: 10.1137/16M1058546.  Google Scholar

[19]

N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056. doi: 10.1073/pnas.51.6.1055.  Google Scholar

[20]

G. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.   Google Scholar

[21]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122.  doi: 10.1007/BF02413623.  Google Scholar

[22]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[23]

G. Riey, Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9.   Google Scholar

[24]

G. Riey and B. Sciunzi, A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858.  doi: 10.1051/cocv/2017040.  Google Scholar

[25]

B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334.  doi: 10.1007/s00030-007-5047-7.  Google Scholar

[26]

B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp. doi: 10.1142/S0219199714500138.  Google Scholar

[27]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[28]

N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.   Google Scholar

[29]

V. V. Zhykov, Averaging of functional of the calculus of variations and elasticity theory, Izk. Akad. Nauk. SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar

[30]

V. V. Zhykov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

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