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February  2019, 12(1): 43-56. doi: 10.3934/dcdss.2019003

Asymptotics for quasilinear obstacle problems in bad domains

Raffaela Capitanelli , and  Salvatore Fragapane

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy

* Corresponding author

Received  March 2017 Revised  October 2017 Published  July 2018

Fund Project: The first author is supported by INdAM GNAMPA Project 2016 and Grant Ateneo "Sapienza" 2015.

We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for $p \to \infty $ and $n \to \infty $.

Keywords: Fractals, degenerate elliptic equations, variational methods for second-order elliptic equations, quasilinear elliptic equations with p-Laplacian, asymptotic behavior of solutions.
Mathematics Subject Classification: Primary: 28A80, 35J70, 35J20, 35J92, 35B40.
Citation: Raffaela Capitanelli, Salvatore Fragapane. Asymptotics for quasilinear obstacle problems in bad domains. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 43-56. doi: 10.3934/dcdss.2019003
References:
[1]

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

[2]

J. W. Barrett and W. B Liu, Finite element approximation of the p-Laplacian, Math. Comp., 61 (1993), 523-537.  doi: 10.2307/2153239.

[3]

T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p \to + \infty $ of ${\Delta _p}{u_p} = f$ and related extremal problems, Some Topics in Nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 15-68 (1991).

[4]

F. Camilli, R. Capitanelli and M. A. Vivaldi, Absolutely minimizing Lipschitz extensions and infinity harmonic functions on the Sierpinski gasket, Nonlinear Anal., 163 (2017), 71-85.  doi: 10.1016/j.na.2017.07.005.

[5]

R. Capitanelli, Asymptotics for mixed Dirichlet-Robin problems in irregular domains, J. Math. Anal. Appl., 362 (2010), 450-459.  doi: 10.1016/j.jmaa.2009.09.042.

[6]

R. Capitanelli and M. A. Vivaldi, Dynamical Quasi-Filling Fractal Layers, SIAM J. Math. Anal., 48 (2016), 3931-3961.  doi: 10.1137/15M1043893.

[7]

R. Capitanelli and M. A. Vivaldi, FEM for quasilinear obstacle problems in bad domains, ESAIM Math. Model. Numer. Anal., 51 (2017), 2465-2485.  doi: 10.1051/m2an/2017033.

[8]

J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. Ⅰ. Elliptic equations. Research Notes in Mathematics. 106. Pitman, Boston, MA, 1985.

[9]

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. doi: 10.1090/memo/0653.

[10]

D. S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces, Phys. Rev. E, 73 (2006), 021103, 9pp. doi: 10.1103/PhysRevE.73.021103.

[11]

J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J, 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.

[12]

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.

[13]

J. M. Mazón, J. D. Rossi and J. Toledo, Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles, Journal of Differential Equations, 256 (2014), 3208-3244.  doi: 10.1016/j.jde.2014.01.039.

[14]

E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842.  doi: 10.1090/S0002-9904-1934-05978-0.

[15]

U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7.

[16]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials, J. Math. Pures Appl. (9), 103 (2015), 1198-1227.  doi: 10.1016/j.matpur.2014.10.010.

[17]

Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[18]

H. L. Royden, Real Analysis, Third edition. Macmillan Publishing Company, New York, 1988.

[19]

G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Springer, 1987. doi: 10.1007/978-1-4899-3614-1.

[20]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

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

[2]

J. W. Barrett and W. B Liu, Finite element approximation of the p-Laplacian, Math. Comp., 61 (1993), 523-537.  doi: 10.2307/2153239.

[3]

T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p \to + \infty $ of ${\Delta _p}{u_p} = f$ and related extremal problems, Some Topics in Nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 15-68 (1991).

[4]

F. Camilli, R. Capitanelli and M. A. Vivaldi, Absolutely minimizing Lipschitz extensions and infinity harmonic functions on the Sierpinski gasket, Nonlinear Anal., 163 (2017), 71-85.  doi: 10.1016/j.na.2017.07.005.

[5]

R. Capitanelli, Asymptotics for mixed Dirichlet-Robin problems in irregular domains, J. Math. Anal. Appl., 362 (2010), 450-459.  doi: 10.1016/j.jmaa.2009.09.042.

[6]

R. Capitanelli and M. A. Vivaldi, Dynamical Quasi-Filling Fractal Layers, SIAM J. Math. Anal., 48 (2016), 3931-3961.  doi: 10.1137/15M1043893.

[7]

R. Capitanelli and M. A. Vivaldi, FEM for quasilinear obstacle problems in bad domains, ESAIM Math. Model. Numer. Anal., 51 (2017), 2465-2485.  doi: 10.1051/m2an/2017033.

[8]

J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. Ⅰ. Elliptic equations. Research Notes in Mathematics. 106. Pitman, Boston, MA, 1985.

[9]

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. doi: 10.1090/memo/0653.

[10]

D. S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces, Phys. Rev. E, 73 (2006), 021103, 9pp. doi: 10.1103/PhysRevE.73.021103.

[11]

J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J, 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.

[12]

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.

[13]

J. M. Mazón, J. D. Rossi and J. Toledo, Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles, Journal of Differential Equations, 256 (2014), 3208-3244.  doi: 10.1016/j.jde.2014.01.039.

[14]

E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837-842.  doi: 10.1090/S0002-9904-1934-05978-0.

[15]

U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7.

[16]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials, J. Math. Pures Appl. (9), 103 (2015), 1198-1227.  doi: 10.1016/j.matpur.2014.10.010.

[17]

Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[18]

H. L. Royden, Real Analysis, Third edition. Macmillan Publishing Company, New York, 1988.

[19]

G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Springer, 1987. doi: 10.1007/978-1-4899-3614-1.

[20]

C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

Figure 1.  $\Omega_3^n$
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Figure 2.  A bad domain
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Figure 3.  Second step
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Figure 4.  Third step
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Figure 5.  Fourth step
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