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Quasilinear elliptic equations with measures and multi-valued lower order terms
A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis
1. | Università degli Studi di Trieste, Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Piazzale Europa 1,34127 Trieste, Italy |
2. | Univ. Valenciennes, EA 4015 -LAMAV -FR CNRS 2956, F-59313 Valenciennes, France |
3. | Università degli Studi di Trieste, Dipartimento di Matematica e Geoscienze-Sezione di Matematica e Informatica, Via A. Valerio 12/1,34127 Trieste, Italy |
$\begin{equation*}{\rm{ -div}}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},\end{equation*}$ |
$Ω \subset \mathbb{R}^N$ |
$a,b>0$ |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems Clarendon Press, Oxford, 2000. |
[2] |
G. Anzellotti,
Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.
doi: 10.1007/BF01781073. |
[3] |
M. Athanassenas and J. Clutterbuck,
A capillarity problem for compressible liquids, Pacific J. Math., 243 (2009), 213-232.
doi: 10.2140/pjm.2009.243.213. |
[4] |
M. Athanassenas and R. Finn,
Compressible fluids in a capillary tube, Pacific J. Math., 224 (2006), 201-229.
doi: 10.2140/pjm.2006.224.201. |
[5] |
M. Bergner,
The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb{R}^{n+1}$, Analysis (Munich), 28 (2008), 149-166.
doi: 10.1524/anly.2008.0906. |
[6] |
M. Bergner,
On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl., 27 (2009), 335-343.
doi: 10.1016/j.difgeo.2009.03.002. |
[7] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari,
Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237.
doi: 10.1016/j.jde.2007.05.031. |
[8] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari,
Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.
|
[9] |
I. Coelho, C. Corsato and P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape Bound. Value Probl. , 2014 (2014), p127.
doi: 10.1186/1687-2770-2014-127. |
[10] |
C. Corsato, C. De Coster and P. Omari,
Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape, Discrete Contin. Dyn. Syst.(Suppl.), 2015 (2015), 297-303.
doi: 10.3934/proc.2015.0297. |
[11] |
C. Corsato, C. De Coster and P. Omari,
The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, J. Differential Equations, 260 (2016), 4572-4618.
doi: 10.1016/j.jde.2015.11.024. |
[12] |
L. Dupaigne,
Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall/CRC, Boca Raton, 2011.
doi: 10.1201/b10802. |
[13] |
I. Ekeland and R. Temam,
Convex Analysis and Variational Problems Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[14] |
L. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions CRC Press, Boca Raton, 1992. |
[15] |
D. G. de Figueiredo,
Lectures on the Ekeland Variational Principle with Applications and Detours Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 81 Springer, Berlin, 1989. |
[16] |
R. Finn,
On the equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151.
doi: 10.1007/PL00000966. |
[17] |
R. Finn,
Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys., 33 (2004), 47-55.
|
[18] |
R. Finn and G. Luli,
On the capillary problem for compressible fluids, J. Math. Fluid Mech., 9 (2007), 87-103.
doi: 10.1007/s00021-005-0203-5. |
[19] |
E. Gagliardo,
Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27 (1957), 284-305.
|
[20] |
C. Gerhardt,
Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335.
|
[21] |
C. Gerhardt,
Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198.
doi: 10.1007/BF01418314. |
[22] |
C. Gerhardt,
On the regularity of solutions to variational problems in $BV(Ω)$, Math. Z., 149 (1976), 281-286.
doi: 10.1007/BF01175590. |
[23] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order Springer, New York, 2001. |
[24] |
E. Giusti,
On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.
doi: 10.1007/BF01393250. |
[25] |
E. Giusti,
Generalized solutions for the mean curvature equation, Pacific J. Math., 88 (1980), 297-321.
doi: 10.2140/pjm.1980.88.297. |
[26] |
E. Giusti,
Minimal Surfaces and Functions of Bounded Variations Birkhäuser, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[27] |
M. Goebel,
On Fréchet-differentiability of Nemytskij operators acting in Hölder spaces, Glasgow Math. J., 33 (1991), 1-5.
doi: 10.1017/S0017089500007965. |
[28] |
K. Hayasida and Y. Ikeda,
Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates, Nonlinear Anal., 67 (2007), 1-25.
doi: 10.1016/j.na.2006.07.016. |
[29] |
K. Hayasida and M. Nakatani,
On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition, Nagoya Math. J., 157 (2000), 177-209.
doi: 10.1017/S0027763000007248. |
[30] |
R. Huff and J. McCuan,
Minimal graphs with discontinuous boundary values, J. Aust. Math. Soc., 86 (2009), 75-95.
doi: 10.1017/S1446788708000335. |
[31] |
H. Jenkins and J. Serrin,
The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187.
doi: 10.1515/crll.1968.229.170. |
[32] |
G. A. Ladyzhenskaya and N. N. Ural'tseva,
Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pur. Appl. Math., 23 (1970), 677-703.
doi: 10.1002/cpa.3160230409. |
[33] |
A. Lichnewsky,
Principe du maximum local et solutions généralisées de problémes du type hypersurfaces minimales, Bull. Soc. Math. France, 102 (1974), 417-433.
|
[34] |
A. Lichnewsky,
Sur le comportement au bord des solutions généralisées du probléme non paramétrique des surfaces minimales, J. Math. Pures Appl., 53 (1974), 397-425.
|
[35] |
A. Lichnewsky,
Solutions généralisées du probléme des surfaces minimales pour des données au bord non bornées, J. Math. Pures Appl., 57 (1978), 231-253.
|
[36] |
A. Lichnewsky and R. Temam,
Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equation, 30 (1978), 340-364.
doi: 10.1016/0022-0396(78)90005-0. |
[37] |
J. López-Gómez, P. Omari and S. Rivetti,
Positive solutions of one-dimensional indefinite capillarity-type problems: A variational approach, J. Differential Equations, 262 (2017), 2335-2392.
doi: 10.1016/j.jde.2016.10.046. |
[38] |
J. López-Gómez, P. Omari and S. Rivetti,
Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal., 155 (2017), 1-51.
doi: 10.1016/j.na.2017.01.007. |
[39] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. |
[40] |
T. Marquardt,
Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z., 264 (2010), 507-511.
doi: 10.1007/s00209-009-0476-0. |
[41] |
M. Miranda,
Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322.
|
[42] |
M. Miranda,
Maximum principles and minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 667-681.
|
[43] |
C. B. Morrey Jr. ,
Multiple Integrals in the Calculus of Variations Springer, New York, 1966. |
[44] |
J. Nečas,
Direct Methods in the Theory of Elliptic Equations Springer, New York, 2012.
doi: 10.1007/978-3-642-10455-8. |
[45] |
F. Obersnel and P. Omari,
Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation, Discrete Contin. Dyn. Syst., 33 (2013), 305-320.
doi: 10.3934/dcds.2013.33.305. |
[46] |
W. Okrasiński and Ł. Płociniczak,
A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl., 13 (2012), 1498-1505.
doi: 10.1016/j.nonrwa.2011.11.014. |
[47] |
W. Okrasiński and Ł. Płociniczak,
Bessel function model of corneal topography, Appl. Math. Comput., 223 (2013), 436-443.
doi: 10.1016/j.amc.2013.07.097. |
[48] |
W. Okrasiński and Ł. Płociniczak,
Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng., 21 (2013), 1090-1097.
doi: 10.1080/17415977.2012.753443. |
[49] |
H. Pan and R. Xing,
Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Ⅱ, Nonlinear Anal., 74 (2011), 3751-3768.
doi: 10.1016/j.na.2011.03.020. |
[50] |
Ł. Płociniczak, G. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41. Google Scholar |
[51] |
Ł. Płociniczak and W. Okrasiński,
Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng., 23 (2015), 443-456.
doi: 10.1080/17415977.2014.922074. |
[52] |
Ł. Płociniczak, W. Okrasiński, J. J. Nieto and O. Domínguez,
On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl., 414 (2014), 461-471.
doi: 10.1016/j.jmaa.2014.01.010. |
[53] |
P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, in Contributions to nonlinear functional analysis (eds. E. H. Zarantonello), Academic Press, (1971), 11-36 |
[54] |
J. Serrin,
The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Soc. Lond. A, 264 (1969), 413-496.
doi: 10.1098/rsta.1969.0033. |
[55] |
R. Temam,
Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Rational Mech. Anal., 44 (1971/72), 121-156.
doi: 10.1007/BF00281813. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems Clarendon Press, Oxford, 2000. |
[2] |
G. Anzellotti,
Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318.
doi: 10.1007/BF01781073. |
[3] |
M. Athanassenas and J. Clutterbuck,
A capillarity problem for compressible liquids, Pacific J. Math., 243 (2009), 213-232.
doi: 10.2140/pjm.2009.243.213. |
[4] |
M. Athanassenas and R. Finn,
Compressible fluids in a capillary tube, Pacific J. Math., 224 (2006), 201-229.
doi: 10.2140/pjm.2006.224.201. |
[5] |
M. Bergner,
The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb{R}^{n+1}$, Analysis (Munich), 28 (2008), 149-166.
doi: 10.1524/anly.2008.0906. |
[6] |
M. Bergner,
On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl., 27 (2009), 335-343.
doi: 10.1016/j.difgeo.2009.03.002. |
[7] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari,
Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237.
doi: 10.1016/j.jde.2007.05.031. |
[8] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari,
Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.
|
[9] |
I. Coelho, C. Corsato and P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape Bound. Value Probl. , 2014 (2014), p127.
doi: 10.1186/1687-2770-2014-127. |
[10] |
C. Corsato, C. De Coster and P. Omari,
Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape, Discrete Contin. Dyn. Syst.(Suppl.), 2015 (2015), 297-303.
doi: 10.3934/proc.2015.0297. |
[11] |
C. Corsato, C. De Coster and P. Omari,
The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, J. Differential Equations, 260 (2016), 4572-4618.
doi: 10.1016/j.jde.2015.11.024. |
[12] |
L. Dupaigne,
Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall/CRC, Boca Raton, 2011.
doi: 10.1201/b10802. |
[13] |
I. Ekeland and R. Temam,
Convex Analysis and Variational Problems Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[14] |
L. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions CRC Press, Boca Raton, 1992. |
[15] |
D. G. de Figueiredo,
Lectures on the Ekeland Variational Principle with Applications and Detours Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 81 Springer, Berlin, 1989. |
[16] |
R. Finn,
On the equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151.
doi: 10.1007/PL00000966. |
[17] |
R. Finn,
Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys., 33 (2004), 47-55.
|
[18] |
R. Finn and G. Luli,
On the capillary problem for compressible fluids, J. Math. Fluid Mech., 9 (2007), 87-103.
doi: 10.1007/s00021-005-0203-5. |
[19] |
E. Gagliardo,
Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27 (1957), 284-305.
|
[20] |
C. Gerhardt,
Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335.
|
[21] |
C. Gerhardt,
Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198.
doi: 10.1007/BF01418314. |
[22] |
C. Gerhardt,
On the regularity of solutions to variational problems in $BV(Ω)$, Math. Z., 149 (1976), 281-286.
doi: 10.1007/BF01175590. |
[23] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order Springer, New York, 2001. |
[24] |
E. Giusti,
On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.
doi: 10.1007/BF01393250. |
[25] |
E. Giusti,
Generalized solutions for the mean curvature equation, Pacific J. Math., 88 (1980), 297-321.
doi: 10.2140/pjm.1980.88.297. |
[26] |
E. Giusti,
Minimal Surfaces and Functions of Bounded Variations Birkhäuser, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[27] |
M. Goebel,
On Fréchet-differentiability of Nemytskij operators acting in Hölder spaces, Glasgow Math. J., 33 (1991), 1-5.
doi: 10.1017/S0017089500007965. |
[28] |
K. Hayasida and Y. Ikeda,
Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates, Nonlinear Anal., 67 (2007), 1-25.
doi: 10.1016/j.na.2006.07.016. |
[29] |
K. Hayasida and M. Nakatani,
On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition, Nagoya Math. J., 157 (2000), 177-209.
doi: 10.1017/S0027763000007248. |
[30] |
R. Huff and J. McCuan,
Minimal graphs with discontinuous boundary values, J. Aust. Math. Soc., 86 (2009), 75-95.
doi: 10.1017/S1446788708000335. |
[31] |
H. Jenkins and J. Serrin,
The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187.
doi: 10.1515/crll.1968.229.170. |
[32] |
G. A. Ladyzhenskaya and N. N. Ural'tseva,
Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pur. Appl. Math., 23 (1970), 677-703.
doi: 10.1002/cpa.3160230409. |
[33] |
A. Lichnewsky,
Principe du maximum local et solutions généralisées de problémes du type hypersurfaces minimales, Bull. Soc. Math. France, 102 (1974), 417-433.
|
[34] |
A. Lichnewsky,
Sur le comportement au bord des solutions généralisées du probléme non paramétrique des surfaces minimales, J. Math. Pures Appl., 53 (1974), 397-425.
|
[35] |
A. Lichnewsky,
Solutions généralisées du probléme des surfaces minimales pour des données au bord non bornées, J. Math. Pures Appl., 57 (1978), 231-253.
|
[36] |
A. Lichnewsky and R. Temam,
Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equation, 30 (1978), 340-364.
doi: 10.1016/0022-0396(78)90005-0. |
[37] |
J. López-Gómez, P. Omari and S. Rivetti,
Positive solutions of one-dimensional indefinite capillarity-type problems: A variational approach, J. Differential Equations, 262 (2017), 2335-2392.
doi: 10.1016/j.jde.2016.10.046. |
[38] |
J. López-Gómez, P. Omari and S. Rivetti,
Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal., 155 (2017), 1-51.
doi: 10.1016/j.na.2017.01.007. |
[39] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. |
[40] |
T. Marquardt,
Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z., 264 (2010), 507-511.
doi: 10.1007/s00209-009-0476-0. |
[41] |
M. Miranda,
Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322.
|
[42] |
M. Miranda,
Maximum principles and minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 667-681.
|
[43] |
C. B. Morrey Jr. ,
Multiple Integrals in the Calculus of Variations Springer, New York, 1966. |
[44] |
J. Nečas,
Direct Methods in the Theory of Elliptic Equations Springer, New York, 2012.
doi: 10.1007/978-3-642-10455-8. |
[45] |
F. Obersnel and P. Omari,
Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation, Discrete Contin. Dyn. Syst., 33 (2013), 305-320.
doi: 10.3934/dcds.2013.33.305. |
[46] |
W. Okrasiński and Ł. Płociniczak,
A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl., 13 (2012), 1498-1505.
doi: 10.1016/j.nonrwa.2011.11.014. |
[47] |
W. Okrasiński and Ł. Płociniczak,
Bessel function model of corneal topography, Appl. Math. Comput., 223 (2013), 436-443.
doi: 10.1016/j.amc.2013.07.097. |
[48] |
W. Okrasiński and Ł. Płociniczak,
Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng., 21 (2013), 1090-1097.
doi: 10.1080/17415977.2012.753443. |
[49] |
H. Pan and R. Xing,
Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Ⅱ, Nonlinear Anal., 74 (2011), 3751-3768.
doi: 10.1016/j.na.2011.03.020. |
[50] |
Ł. Płociniczak, G. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41. Google Scholar |
[51] |
Ł. Płociniczak and W. Okrasiński,
Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng., 23 (2015), 443-456.
doi: 10.1080/17415977.2014.922074. |
[52] |
Ł. Płociniczak, W. Okrasiński, J. J. Nieto and O. Domínguez,
On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl., 414 (2014), 461-471.
doi: 10.1016/j.jmaa.2014.01.010. |
[53] |
P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, in Contributions to nonlinear functional analysis (eds. E. H. Zarantonello), Academic Press, (1971), 11-36 |
[54] |
J. Serrin,
The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Soc. Lond. A, 264 (1969), 413-496.
doi: 10.1098/rsta.1969.0033. |
[55] |
R. Temam,
Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Rational Mech. Anal., 44 (1971/72), 121-156.
doi: 10.1007/BF00281813. |





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