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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[32]

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

[1]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure & Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163

[2]

Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133

[3]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033

[4]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69

[5]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747

[6]

Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift. Communications on Pure & Applied Analysis, 2015, 14 (2) : 407-419. doi: 10.3934/cpaa.2015.14.407

[7]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

[8]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747

[9]

Byungsoo Kang, Hyunseok Kim. W1, p-estimates for elliptic equations with lower order terms. Communications on Pure & Applied Analysis, 2017, 16 (3) : 799-822. doi: 10.3934/cpaa.2017038

[10]

Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743

[11]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[12]

Kim Knudsen, Mikko Salo. Determining nonsmooth first order terms from partial boundary measurements. Inverse Problems & Imaging, 2007, 1 (2) : 349-369. doi: 10.3934/ipi.2007.1.349

[13]

Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507

[14]

Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489

[15]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477

[16]

Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure & Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285

[17]

Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141

[18]

Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935

[19]

Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012

[20]

Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

[Back to Top]