May  2012, 11(3): 1217-1229. doi: 10.3934/cpaa.2012.11.1217

Limits of anisotropic and degenerate elliptic problems

1. 

CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal, Portugal

2. 

Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Received  December 2010 Revised  February 2011 Published  December 2011

This paper analyzes the behavior of solutions for anisotropic problems of $(p_i)$-Laplacian type as the exponents go to infinity. We show that solutions converge uniformly to a function that solves, in the viscosity sense, a certain problem that we identify. The results are presented in a two-dimensional setting but can be extended to any dimension.
Citation: Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217
References:
[1]

G. Aronsson, Extensions of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.  Google Scholar

[2]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[3]

E. N. Barron, L. C. Evans and R. Jensen, The infinity laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3.  Google Scholar

[4]

M. Belloni and B. Kawohl, The pseudo $p$-Laplace eigenvalue problem and viscosity solutions as $p \rightarrow \infty$, ESAIM COCV, 10 (2004), 28-52. doi: 10.1051/cocv:2003035.  Google Scholar

[5]

T. Bhattacharya, E. DiBenedetto and J. J. Manfredi, Limits as $p\rightarrow \infty$ of $\Delta_p u_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1989), 15-68.  Google Scholar

[6]

L. Boccardo, P. Marcellini and C. Sbordone, $L^\infty$-regularity for variational problems with sharp non-standard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.  Google Scholar

[7]

M. G. Crandall, A Visit with the $\infty$-Laplace Equation, in "Calculus of Variations and Nonlinear Partial Differential Equations" (C.I.M.E. Summer School, Cetraro, 2005), Lecture Notes in Math, vol. 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3.  Google Scholar

[8]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[9]

A. Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlin. Stud., 9 (2009), 367-393.  Google Scholar

[10]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), viii+66 pp.  Google Scholar

[11]

I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéeaire, 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001.  Google Scholar

[12]

J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, The Neumann problem for the $\infty$-Laplacian and the Monge-Kantorovich mass transfer problem, Nonlinear Anal., 66 (2007), 349-366. doi: 10.1016/j.na.2005.11.030.  Google Scholar

[13]

J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, Limits for Monge-Kantorovich mass transport problems, Commun. Pure Appl. Anal., 7 (2008), 853-865. doi: 10.3934/cpaa.2008.7.853.  Google Scholar

[14]

T. Ishibashi and S. Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms, SIAM J. Math. Anal., 33 (2001), 545-569. doi: 10.1137/S0036141000380000.  Google Scholar

[15]

H. Ishii, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type, Proc. Amer. Math. Soc., 100 (1987), 247-251. doi: 10.1090/S0002-9939-1987-0884461-3.  Google Scholar

[16]

H. Ishii and P. Loreti, Limits of solutions of $p$-Laplace equations as $p$ goes to infinity and related variational problems, SIAM J. Math. Anal., 37 (2005), 411-437. doi: 10.1137/S0036141004432827.  Google Scholar

[17]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar

[18]

P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions, Dissertation, University of Jyväskulä in Jyväskulä, 1998., Ann. Acad. Sci. Fenn. Math. Diss., 115 (1998), 53 pp.  Google Scholar

[19]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar

[20]

S. N. Kruzhkov and I. M. Kolodii, On the theory of anisotropic Sobolev spaces, Russian Math. Surveys, 38 (1983), 188-189. doi: 10.1070/RM1983v038n02ABEH003476.  Google Scholar

[21]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. doi: 10.1007/978-3-642-11030-6_1.  Google Scholar

[22]

J. J. Manfredi, J. D. Rossi and J. M. Urbano, $p(x)$-Harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595. doi: 10.1016/j.anihpc.2009.09.008.  Google Scholar

[23]

J. J. Manfredi, J. D. Rossi and J. M. Urbano, Limit as $p(x)\rightarrow \infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315. doi: 10.1016/j.na.2009.06.054.  Google Scholar

[24]

S. M. Nikolskii, An imbedding theorem for functions with partial derivatives considered in different metrics, Izd. Akad. Nauk SSSR Ser. Mat., 22 (1958), 321-336.  Google Scholar

[25]

Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048.  Google Scholar

[26]

M. Pérez-Llanos and J. D. Rossi, An anisotropic infinity laplacian obtained as the limit of the anisotropic $(p,q)$-Laplacian, Commun. Contemp. Math., 13 (2011), 1-20. Google Scholar

[27]

M. Pérez-Llanos and J. D. Rossi, The limit as $p(x) \rightarrow +\infty$ of solutions to the inhomogeneous Dirichlet problem of the $p(x)$-Laplacian, Nonlinear Anal., 73 (2010), 2027-2035. doi: 10.1016/j.na.2010.05.032.  Google Scholar

[28]

M. Pérez-Llanos and J. D. Rossi, Limits as $p(x) \to \infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Contemporary Mathematics, 540 (2011), 187-201. Google Scholar

[29]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.  Google Scholar

[30]

M. Troisi, Ulteriori contributi alla teoria degli spazi di Sobolev non isotropi, Ricerche Mat., 20 (1971), 90-117.  Google Scholar

show all references

References:
[1]

G. Aronsson, Extensions of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.  Google Scholar

[2]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[3]

E. N. Barron, L. C. Evans and R. Jensen, The infinity laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3.  Google Scholar

[4]

M. Belloni and B. Kawohl, The pseudo $p$-Laplace eigenvalue problem and viscosity solutions as $p \rightarrow \infty$, ESAIM COCV, 10 (2004), 28-52. doi: 10.1051/cocv:2003035.  Google Scholar

[5]

T. Bhattacharya, E. DiBenedetto and J. J. Manfredi, Limits as $p\rightarrow \infty$ of $\Delta_p u_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1989), 15-68.  Google Scholar

[6]

L. Boccardo, P. Marcellini and C. Sbordone, $L^\infty$-regularity for variational problems with sharp non-standard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.  Google Scholar

[7]

M. G. Crandall, A Visit with the $\infty$-Laplace Equation, in "Calculus of Variations and Nonlinear Partial Differential Equations" (C.I.M.E. Summer School, Cetraro, 2005), Lecture Notes in Math, vol. 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3.  Google Scholar

[8]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[9]

A. Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlin. Stud., 9 (2009), 367-393.  Google Scholar

[10]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), viii+66 pp.  Google Scholar

[11]

I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéeaire, 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001.  Google Scholar

[12]

J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, The Neumann problem for the $\infty$-Laplacian and the Monge-Kantorovich mass transfer problem, Nonlinear Anal., 66 (2007), 349-366. doi: 10.1016/j.na.2005.11.030.  Google Scholar

[13]

J. García-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, Limits for Monge-Kantorovich mass transport problems, Commun. Pure Appl. Anal., 7 (2008), 853-865. doi: 10.3934/cpaa.2008.7.853.  Google Scholar

[14]

T. Ishibashi and S. Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms, SIAM J. Math. Anal., 33 (2001), 545-569. doi: 10.1137/S0036141000380000.  Google Scholar

[15]

H. Ishii, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type, Proc. Amer. Math. Soc., 100 (1987), 247-251. doi: 10.1090/S0002-9939-1987-0884461-3.  Google Scholar

[16]

H. Ishii and P. Loreti, Limits of solutions of $p$-Laplace equations as $p$ goes to infinity and related variational problems, SIAM J. Math. Anal., 37 (2005), 411-437. doi: 10.1137/S0036141004432827.  Google Scholar

[17]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar

[18]

P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions, Dissertation, University of Jyväskulä in Jyväskulä, 1998., Ann. Acad. Sci. Fenn. Math. Diss., 115 (1998), 53 pp.  Google Scholar

[19]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar

[20]

S. N. Kruzhkov and I. M. Kolodii, On the theory of anisotropic Sobolev spaces, Russian Math. Surveys, 38 (1983), 188-189. doi: 10.1070/RM1983v038n02ABEH003476.  Google Scholar

[21]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. doi: 10.1007/978-3-642-11030-6_1.  Google Scholar

[22]

J. J. Manfredi, J. D. Rossi and J. M. Urbano, $p(x)$-Harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595. doi: 10.1016/j.anihpc.2009.09.008.  Google Scholar

[23]

J. J. Manfredi, J. D. Rossi and J. M. Urbano, Limit as $p(x)\rightarrow \infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315. doi: 10.1016/j.na.2009.06.054.  Google Scholar

[24]

S. M. Nikolskii, An imbedding theorem for functions with partial derivatives considered in different metrics, Izd. Akad. Nauk SSSR Ser. Mat., 22 (1958), 321-336.  Google Scholar

[25]

Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048.  Google Scholar

[26]

M. Pérez-Llanos and J. D. Rossi, An anisotropic infinity laplacian obtained as the limit of the anisotropic $(p,q)$-Laplacian, Commun. Contemp. Math., 13 (2011), 1-20. Google Scholar

[27]

M. Pérez-Llanos and J. D. Rossi, The limit as $p(x) \rightarrow +\infty$ of solutions to the inhomogeneous Dirichlet problem of the $p(x)$-Laplacian, Nonlinear Anal., 73 (2010), 2027-2035. doi: 10.1016/j.na.2010.05.032.  Google Scholar

[28]

M. Pérez-Llanos and J. D. Rossi, Limits as $p(x) \to \infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Contemporary Mathematics, 540 (2011), 187-201. Google Scholar

[29]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.  Google Scholar

[30]

M. Troisi, Ulteriori contributi alla teoria degli spazi di Sobolev non isotropi, Ricerche Mat., 20 (1971), 90-117.  Google Scholar

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