April 2018, 11(2): 213-256. doi: 10.3934/dcdss.2018013

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

* Corresponding author: Pierpaolo Omari

Received  December 2016 Revised  April 2017 Published  January 2018

Fund Project: This paper was written under the auspices of INdAM-GNAMPA. The third and the fourth named authors have also been supported by the University of Trieste, in the frame of the 2015 FRA project "Differential Equations: Qualitative and Computational Theory"

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation
$\begin{equation*}{\rm{ -div}}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},\end{equation*}$
in a bounded Lipschitz domain
$Ω \subset \mathbb{R}^N$
, with
$a,b>0$
parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.
Citation: Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013
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. BonheureP. HabetsF. 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. BonheureP. HabetsF. 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. CorsatoC. 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. CorsatoC. 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.

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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.

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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.

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R. Finn, On the equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151. doi: 10.1007/PL00000966.

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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-GómezP. 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-GómezP. 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. PłociniczakG. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41.

[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. PłociniczakW. OkrasińskiJ. 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. BonheureP. HabetsF. 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. BonheureP. HabetsF. 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. CorsatoC. 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. CorsatoC. 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-GómezP. 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-GómezP. 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. PłociniczakG. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41.

[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. PłociniczakW. OkrasińskiJ. 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.

Figure 1.  Graph of a generalized solution on an arbitrary domain.
Figure 2.  Graph of a singular solution on a thick spherical shell
Figure 3.  Classical solutions emanating from the trivial line: $\|\nabla u (a, b)\|_\infty$ is plotted, in applicates, versus $a$, in abscissas, and $b$, in ordinates
Figure 4.  Profile of the upper solution
Figure 5.  Profile of the lower solution
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Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319

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Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

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Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225

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Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443

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