May  2015, 14(3): 897-922. doi: 10.3934/cpaa.2015.14.897

Gradient estimates and comparison principle for some nonlinear elliptic equations

1. 

Università degli Studi di Napoli "Parthenope", Dipartimento di Ingegneria, Centro Direzionale, Isola C4 80143 Napoli, Italy

2. 

Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy, Italy

3. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli

Received  July 2014 Revised  January 2015 Published  March 2015

We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators.
Citation: Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
References:
[1]

A. Alvino, Sharp a priori estimates for some nonlinear elliptic problems,, Boll. Accademia Gioenia di Scienze Naturali in Catania, 46 (2013), 2. Google Scholar

[2]

A. Alvino, M. F. Betta and A. Mercaldo, Comparison principle for some class of nonlinear elliptic equations,, J. Differential Equations, 12 (2010), 3279. doi: 10.1016/j.jde.2010.07.030. Google Scholar

[3]

A. Alvino, A. Cianchi, V. G. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017. doi: 10.1016/j.anihpc.2010.01.010. Google Scholar

[4]

A. Alvino, V. Ferone and A. Mercaldo, Sharp a-priori estimates for a class of nonlinear elliptic equations with lower order terms,, Ann. Mat. Pura Appl., (): 10231. Google Scholar

[5]

A. Alvino, V. Ferone and G. Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with $L^1$ data,, Ann. Mat. Pura Appl., 178 (2000), 129. doi: 10.1007/BF02505892. Google Scholar

[6]

A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^{1}$ data: an approach via symmetrization methods,, Mediterr. J. Math, 5 (2008), 173. doi: 10.1007/s00009-008-0142-5. Google Scholar

[7]

A. Alvino and A. Mercaldo, Nonlinear elliptic equations with lower order terms and symmetrization methods,, Boll Unione Mat. Ital., 1 (2008), 645. Google Scholar

[8]

G. Barles, G. Díaz and J. I. Díaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity,, Comm. Partial Differential Equations, 17 (1992), 1037. doi: 10.1080/03605309208820876. Google Scholar

[9]

G. Barles and A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations,, Ann. Scuola Norm. Sup., 5 (2006), 107. Google Scholar

[10]

Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^{1}$ theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 22 (1995), 241. Google Scholar

[11]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar

[12]

M. F. Betta and A. Mercaldo, Uniqueness results for nonlinear elliptic equations via symmetrization methods,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 21 (2010), 1. doi: 10.4171/RLM/557. Google Scholar

[13]

M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum,, C. R. Math. Acad. Sci. Paris, 334 (2002), 757. doi: 10.1016/S1631-073X(02)02338-5. Google Scholar

[14]

M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with lower-order term and right-hand side in $L^1(\Omega)$, A tribute to J.-L. Lions. (electronic),, ESAIM Control Optim. Calc. Var. 8 (2002), 8 (2002), 239. doi: 10.1051/cocv:2002051. Google Scholar

[15]

M. F. Betta, A. Mercaldo and R. Volpicelli, Continuous dependence on the data for solutions to nonlinear elliptic equations with a lower order term,, Ricerche Mat., 63 (2014), 41. doi: 10.1007/s11587-014-0198-4. Google Scholar

[16]

A. Cianchi and V. G. Maz'ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems,, J. Eur. Math. Soc. (JEMS), 16 (2014), 571. doi: 10.4171/JEMS/440. Google Scholar

[17]

A. Dall'Aglio, Approximated solutions of equations with $L^{1}$ data. Application to the $H$-convergence of quasi-linear parabolic equations,, Ann. Mat. Pura Appl., 170 (1996), 207. doi: 10.1007/BF01758989. Google Scholar

[18]

G. Dal Maso and A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 375. Google Scholar

[19]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741. Google Scholar

[20]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., 133 (2011), 1093. doi: 10.1353/ajm.2011.0023. Google Scholar

[21]

V. Ferone and B. Messano, Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient,, Advanced Nonlinear Studies, 7 (2007), 31. Google Scholar

[22]

V. Ferone and F. Murat, Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces,, J. Differential Equations, 256 (2014), 577. doi: 10.1016/j.jde.2013.09.013. Google Scholar

[23]

N. Grenon, F. Murat and A. Porretta, A priori estimates and existence for elliptic equations with gradient dependent term,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (2014), 137. Google Scholar

[24]

O. Guibé and A. Mercaldo, Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms,, Commun. Pure Appl. Anal., 7 (2008), 163. Google Scholar

[25]

R. Hunt, On L(p,q) spaces,, Enseignement Math., 12 (1966), 249. Google Scholar

[26]

B. Kawohl, Rearrangements and Convexity of Level Sets in P.D.E.,, Lecture Notes in Mathematics, 1150 (1985). Google Scholar

[27]

J. Leray and J.-L. Lions, Quelques résulatats de Visik sur les problées elliptiques non linéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97. Google Scholar

[28]

P.- L. Lions and F. Murat, Sur les solutions renormalisées d'equations elliptiques non linéaires,, manuscript., (). Google Scholar

[29]

A. Mercaldo, A priori estimates and comparison principle for some nonlinear elliptic equations,, in Geometric Properties for Parabolics and Elliptic PDE's. Springer INdAM Series, (2013), 223. doi: 10.1007/978-88-470-2841-8_14. Google Scholar

[30]

F. Murat, Soluciones renormalizadas de EDP elipticas no lineales,, Preprint 93023, (1993). Google Scholar

[31]

A. Porretta, On the comparison principle for p-laplace operators with first order terms,, in On the notions of solution to nonlinear elliptic problems: results and developments, (2008), 459. Google Scholar

[32]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,, Ann. Mat. Pura Appl., 120 (1979), 160. doi: 10.1007/BF02411942. Google Scholar

show all references

References:
[1]

A. Alvino, Sharp a priori estimates for some nonlinear elliptic problems,, Boll. Accademia Gioenia di Scienze Naturali in Catania, 46 (2013), 2. Google Scholar

[2]

A. Alvino, M. F. Betta and A. Mercaldo, Comparison principle for some class of nonlinear elliptic equations,, J. Differential Equations, 12 (2010), 3279. doi: 10.1016/j.jde.2010.07.030. Google Scholar

[3]

A. Alvino, A. Cianchi, V. G. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017. doi: 10.1016/j.anihpc.2010.01.010. Google Scholar

[4]

A. Alvino, V. Ferone and A. Mercaldo, Sharp a-priori estimates for a class of nonlinear elliptic equations with lower order terms,, Ann. Mat. Pura Appl., (): 10231. Google Scholar

[5]

A. Alvino, V. Ferone and G. Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with $L^1$ data,, Ann. Mat. Pura Appl., 178 (2000), 129. doi: 10.1007/BF02505892. Google Scholar

[6]

A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^{1}$ data: an approach via symmetrization methods,, Mediterr. J. Math, 5 (2008), 173. doi: 10.1007/s00009-008-0142-5. Google Scholar

[7]

A. Alvino and A. Mercaldo, Nonlinear elliptic equations with lower order terms and symmetrization methods,, Boll Unione Mat. Ital., 1 (2008), 645. Google Scholar

[8]

G. Barles, G. Díaz and J. I. Díaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity,, Comm. Partial Differential Equations, 17 (1992), 1037. doi: 10.1080/03605309208820876. Google Scholar

[9]

G. Barles and A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations,, Ann. Scuola Norm. Sup., 5 (2006), 107. Google Scholar

[10]

Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^{1}$ theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 22 (1995), 241. Google Scholar

[11]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar

[12]

M. F. Betta and A. Mercaldo, Uniqueness results for nonlinear elliptic equations via symmetrization methods,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 21 (2010), 1. doi: 10.4171/RLM/557. Google Scholar

[13]

M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum,, C. R. Math. Acad. Sci. Paris, 334 (2002), 757. doi: 10.1016/S1631-073X(02)02338-5. Google Scholar

[14]

M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with lower-order term and right-hand side in $L^1(\Omega)$, A tribute to J.-L. Lions. (electronic),, ESAIM Control Optim. Calc. Var. 8 (2002), 8 (2002), 239. doi: 10.1051/cocv:2002051. Google Scholar

[15]

M. F. Betta, A. Mercaldo and R. Volpicelli, Continuous dependence on the data for solutions to nonlinear elliptic equations with a lower order term,, Ricerche Mat., 63 (2014), 41. doi: 10.1007/s11587-014-0198-4. Google Scholar

[16]

A. Cianchi and V. G. Maz'ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems,, J. Eur. Math. Soc. (JEMS), 16 (2014), 571. doi: 10.4171/JEMS/440. Google Scholar

[17]

A. Dall'Aglio, Approximated solutions of equations with $L^{1}$ data. Application to the $H$-convergence of quasi-linear parabolic equations,, Ann. Mat. Pura Appl., 170 (1996), 207. doi: 10.1007/BF01758989. Google Scholar

[18]

G. Dal Maso and A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 375. Google Scholar

[19]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741. Google Scholar

[20]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., 133 (2011), 1093. doi: 10.1353/ajm.2011.0023. Google Scholar

[21]

V. Ferone and B. Messano, Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient,, Advanced Nonlinear Studies, 7 (2007), 31. Google Scholar

[22]

V. Ferone and F. Murat, Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces,, J. Differential Equations, 256 (2014), 577. doi: 10.1016/j.jde.2013.09.013. Google Scholar

[23]

N. Grenon, F. Murat and A. Porretta, A priori estimates and existence for elliptic equations with gradient dependent term,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (2014), 137. Google Scholar

[24]

O. Guibé and A. Mercaldo, Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms,, Commun. Pure Appl. Anal., 7 (2008), 163. Google Scholar

[25]

R. Hunt, On L(p,q) spaces,, Enseignement Math., 12 (1966), 249. Google Scholar

[26]

B. Kawohl, Rearrangements and Convexity of Level Sets in P.D.E.,, Lecture Notes in Mathematics, 1150 (1985). Google Scholar

[27]

J. Leray and J.-L. Lions, Quelques résulatats de Visik sur les problées elliptiques non linéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97. Google Scholar

[28]

P.- L. Lions and F. Murat, Sur les solutions renormalisées d'equations elliptiques non linéaires,, manuscript., (). Google Scholar

[29]

A. Mercaldo, A priori estimates and comparison principle for some nonlinear elliptic equations,, in Geometric Properties for Parabolics and Elliptic PDE's. Springer INdAM Series, (2013), 223. doi: 10.1007/978-88-470-2841-8_14. Google Scholar

[30]

F. Murat, Soluciones renormalizadas de EDP elipticas no lineales,, Preprint 93023, (1993). Google Scholar

[31]

A. Porretta, On the comparison principle for p-laplace operators with first order terms,, in On the notions of solution to nonlinear elliptic problems: results and developments, (2008), 459. Google Scholar

[32]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,, Ann. Mat. Pura Appl., 120 (1979), 160. doi: 10.1007/BF02411942. Google Scholar

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