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.  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.  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.  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.  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.   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.   Google Scholar

[7]

M. G. Crandall, A Visit with the $\infty$-Laplace Equation,, in, (2005).  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.  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.   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).   Google Scholar

[11]

I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 21 (2004), 715.  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.  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.  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.  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.  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.  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.  doi: 10.1007/BF00386368.  Google Scholar

[18]

P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions,, Dissertation, 115 (1998).   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.  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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 2581.  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.  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.   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.  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.   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.  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.   Google Scholar

[29]

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

[30]

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

show all references

References:
[1]

G. Aronsson, Extensions of functions satisfying Lipschitz conditions,, Ark. Mat., 6 (1967), 551.  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.  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.  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.  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.   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.   Google Scholar

[7]

M. G. Crandall, A Visit with the $\infty$-Laplace Equation,, in, (2005).  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.  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.   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).   Google Scholar

[11]

I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 21 (2004), 715.  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.  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.  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.  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.  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.  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.  doi: 10.1007/BF00386368.  Google Scholar

[18]

P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions,, Dissertation, 115 (1998).   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.  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.  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.  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. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 2581.  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.  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.   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.  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.   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.  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.   Google Scholar

[29]

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

[30]

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

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