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Uniqueness for Neumann problems for nonlinear elliptic equations
1. | Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola C4 80143 Napoli, Italy |
2. | Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP.12 76801 Saint-Étienne-du-Rouvray, France |
3. | Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy |
$\left\{ \begin{align} & -\text{div}({{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u)-\text{div}(c(x)|u{{|}^{p-2}}u)=f\ \ \ \text{in}\ \Omega , \\ & \left( {{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u+c(x)|u{{|}^{p-2}}u \right)\cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}=0\ \ \ \text{on}\ \partial \Omega , \\ \end{align} \right.$ |
$Ω$ |
$\mathbb{R}^{N}$ |
$N≥ 2$ |
$ 1 < p < N$ |
$\underline n$ |
$\partial Ω$ |
$f$ |
$L^{(p^{*})'}(Ω)$ |
$L^{1}(Ω)$ |
$\int{{}}_Ω f \, dx = 0$ |
$c(x)$ |
References:
[1] |
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), 1017-1054.
doi: 10.1016/j.anihpc.2010.01.010. |
[2] |
A. Alvino and A. Mercaldo,
Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediterr. J. Math., 5 (2008), 173-185.
doi: 10.1007/s00009-008-0142-5. |
[3] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo,
Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1997), 183-213.
|
[4] |
M. Artola,
Sur une classe de problémes paraboliques quasi-linéaires, Boll. Un. Mat. Ital. B (6), 5 (1986), 51-70.
|
[5] |
G. Barles, G. Diaz and J. I. Diaz,
Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity, Comm. Partial Differential
Equations, 17 (1992), 1037-1050.
doi: 10.1080/03605309208820876. |
[6] |
M. Ben Cheikh Ali and O. Guibé,
Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl., 16 (2006), 275-297.
|
[7] |
P. Bénilan, L. Boccardo, T. 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. (4), 22 (1995), 241-273.
|
[8] |
M. F. Betta, O. Guibé and A. Mercaldo,
Neumann problems for nonlinear elliptic equations with $L^1$ data, J. Differential Equations, 259 (2015), 898-924.
doi: 10.1016/j.jde.2015.02.031. |
[9] |
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), 757-762.
doi: 10.1016/S1631-073X(02)02338-5. |
[10] |
M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Existence of renormalized solutions to
nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Math.
Pures Appl. (9), 82 (2003), 90–124. Corrected reprint of J. Math. Pures Appl. (9), 8 (2002),
533–566.
doi: 10.1016/S0021-7824(03)00006-0. |
[11] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.
doi: 10.1016/0022-1236(89)90005-0. |
[12] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[13] |
L. Boccardo, T. Gallouët and F. Murat,
Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris S´er. I Math., 315 (1992), 1159-1164.
|
[14] |
J. Chabrowski,
On the Neumann problem with $L^1$ data, Colloq. Math., 107 (2007), 301-316.
doi: 10.4064/cm107-2-10. |
[15] |
M. Chipot and G. Michaille,
Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 137-166.
|
[16] |
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. (4), 28 (1999), 741-808.
|
[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. (4), 170 (1996), 207-240.
doi: 10.1007/BF01758989. |
[18] |
A. Decarreau, J. Liang and J.-M. Rakotoson,
Trace imbeddings for $T$-sets and application to Neumann-Dirichlet problems with measures included in the boundary data, Ann. Fac. Sci.
Toulouse Math. (6), 5 (1996), 443-470.
|
[19] |
J. Droniou,
Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method, Adv. Differential Equations, 5 (2000), 1341-1396.
|
[20] |
J. Droniou and J.-L. Vázquez,
Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.
doi: 10.1007/s00526-008-0189-y. |
[21] |
V. Ferone and A. Mercaldo,
A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.
doi: 10.1016/S0764-4442(98)85005-2. |
[22] |
V. Ferone and A. Mercaldo,
Neumann problems and Steiner symmetrization, Comm. Partial
Differential Equations, 30 (2005), 1537-1553.
doi: 10.1080/03605300500299596. |
[23] |
O. Guibé and A. Mercaldo,
Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258.
doi: 10.1007/s11118-006-9011-7. |
[24] |
O. Guibé and A. Mercaldo,
Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data, Trans. Amer. Math. Soc., 360 (2008), 643-669.
doi: 10.1090/S0002-9947-07-04139-6. |
[25] |
J. Leray and J.-L. Lions,
Quelques résulatats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
|
[26] |
J.-L. Lions,
Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, 1969. |
[27] |
P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, In manuscript. |
[28] |
F. Murat, Equations elliptiques non linéaires avec second membre ${L}^1$ ou mesure, In Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994. |
[29] |
A. Prignet,
Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Ann. Fac. Sci. Toulouse Math. (6), 6 (1997), 297-318.
|
[30] |
W. P. Ziemer,
Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
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), 1017-1054.
doi: 10.1016/j.anihpc.2010.01.010. |
[2] |
A. Alvino and A. Mercaldo,
Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediterr. J. Math., 5 (2008), 173-185.
doi: 10.1007/s00009-008-0142-5. |
[3] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo,
Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1997), 183-213.
|
[4] |
M. Artola,
Sur une classe de problémes paraboliques quasi-linéaires, Boll. Un. Mat. Ital. B (6), 5 (1986), 51-70.
|
[5] |
G. Barles, G. Diaz and J. I. Diaz,
Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity, Comm. Partial Differential
Equations, 17 (1992), 1037-1050.
doi: 10.1080/03605309208820876. |
[6] |
M. Ben Cheikh Ali and O. Guibé,
Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl., 16 (2006), 275-297.
|
[7] |
P. Bénilan, L. Boccardo, T. 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. (4), 22 (1995), 241-273.
|
[8] |
M. F. Betta, O. Guibé and A. Mercaldo,
Neumann problems for nonlinear elliptic equations with $L^1$ data, J. Differential Equations, 259 (2015), 898-924.
doi: 10.1016/j.jde.2015.02.031. |
[9] |
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), 757-762.
doi: 10.1016/S1631-073X(02)02338-5. |
[10] |
M. F. Betta, A. Mercaldo, F. Murat and M. M. Porzio, Existence of renormalized solutions to
nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Math.
Pures Appl. (9), 82 (2003), 90–124. Corrected reprint of J. Math. Pures Appl. (9), 8 (2002),
533–566.
doi: 10.1016/S0021-7824(03)00006-0. |
[11] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.
doi: 10.1016/0022-1236(89)90005-0. |
[12] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[13] |
L. Boccardo, T. Gallouët and F. Murat,
Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris S´er. I Math., 315 (1992), 1159-1164.
|
[14] |
J. Chabrowski,
On the Neumann problem with $L^1$ data, Colloq. Math., 107 (2007), 301-316.
doi: 10.4064/cm107-2-10. |
[15] |
M. Chipot and G. Michaille,
Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 137-166.
|
[16] |
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. (4), 28 (1999), 741-808.
|
[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. (4), 170 (1996), 207-240.
doi: 10.1007/BF01758989. |
[18] |
A. Decarreau, J. Liang and J.-M. Rakotoson,
Trace imbeddings for $T$-sets and application to Neumann-Dirichlet problems with measures included in the boundary data, Ann. Fac. Sci.
Toulouse Math. (6), 5 (1996), 443-470.
|
[19] |
J. Droniou,
Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method, Adv. Differential Equations, 5 (2000), 1341-1396.
|
[20] |
J. Droniou and J.-L. Vázquez,
Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.
doi: 10.1007/s00526-008-0189-y. |
[21] |
V. Ferone and A. Mercaldo,
A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.
doi: 10.1016/S0764-4442(98)85005-2. |
[22] |
V. Ferone and A. Mercaldo,
Neumann problems and Steiner symmetrization, Comm. Partial
Differential Equations, 30 (2005), 1537-1553.
doi: 10.1080/03605300500299596. |
[23] |
O. Guibé and A. Mercaldo,
Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258.
doi: 10.1007/s11118-006-9011-7. |
[24] |
O. Guibé and A. Mercaldo,
Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data, Trans. Amer. Math. Soc., 360 (2008), 643-669.
doi: 10.1090/S0002-9947-07-04139-6. |
[25] |
J. Leray and J.-L. Lions,
Quelques résulatats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
|
[26] |
J.-L. Lions,
Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, 1969. |
[27] |
P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, In manuscript. |
[28] |
F. Murat, Equations elliptiques non linéaires avec second membre ${L}^1$ ou mesure, In Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994. |
[29] |
A. Prignet,
Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Ann. Fac. Sci. Toulouse Math. (6), 6 (1997), 297-318.
|
[30] |
W. P. Ziemer,
Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation.
doi: 10.1007/978-1-4612-1015-3. |
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