# American Institute of Mathematical Sciences

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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