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September  2013, 12(5): 2267-2275. doi: 10.3934/cpaa.2013.12.2267

Convexity of solutions to boundary blow-up problems

Petri Juutinen 1,

1. 

Department of Mathematics and Statistics, P.O.Box 35, FIN-40014 University of Jyväskylä, Finland

Received  September 2009 Revised  February 2010 Published  January 2013

We prove convexity of solutions to boundary blow-up problems for the singular infinity Laplacian and the $p$-Laplacian for $p\ge 2$. The proof is based on an extension of the results of Alvarez, Lasry and Lions [2] and on estimates of the boundary blow-up rate.
Keywords: boundary blow-up, infinity Laplacian., Convexity.
Mathematics Subject Classification: 35J60, 35J65, 26B2.
Citation: Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
References:
[1]

H. Aikawa, T. Kilpeläinen, N. Shanmugalingam and X. Zhong, Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains, Potential Anal., 26 (2007), 281-301. doi: 10.1007/s11118-006-9036-y.  Google Scholar

[2]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[3]

L. Bieberbach, $\Delta u= e^u$ und die automorphen Funktionen, Math. Ann., 77 (1916), 173-212. doi: 10.1007/BF01456901.  Google Scholar

[4]

O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: asymptotics, uniqueness and symmetry, J. Differential Equations, 249 (2010), 931-964. doi: 10.1016/j.jde.2010.02.023.  Google Scholar

[5]

M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations, 3 (1990), 1001-1014.  Google Scholar

[6]

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

[7]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125. doi: 10.1016/0362-546X(93)90012-H.  Google Scholar

[8]

W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc., 54 (1987), 141-175. doi: 10.1112/plms/s3-54.1.141.  Google Scholar

[9]

M. Feldman, Variational evolution problems and nonlocal geometric motion, Arch. Ration. Mech. Anal., 146 (1999), 221-274. doi: 10.1007/s002050050142.  Google Scholar

[10]

Y. Giga, "Surface Evolution Equations. A Level Set Approach," Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[11]

P. Juutinen, The boundary Harnack inequality for infinite harmonic functions in Lipschitz domains satisfying the interior ball condition, Nonlinear Anal., 69 (2008), 1941-1944. doi: 10.1016/j.na.2007.07.035.  Google Scholar

[12]

P. Juutinen and J. D. Rossi, Large solutions for the infinity Laplacian, Adv. Calc. Var., 1 (2008), 271-289. doi: 10.1515/ACV.2008.011.  Google Scholar

[13]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985.  Google Scholar

[14]

J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar

[15]

Y. Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math., 58 (2005), 85-146. doi: 10.1002/cpa.20051.  Google Scholar

[16]

R. Osserman, On the inequality $\Delta u\ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647. doi: 10.2140/pjm.1957.7.1641.  Google Scholar

show all references

References:
[1]

H. Aikawa, T. Kilpeläinen, N. Shanmugalingam and X. Zhong, Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains, Potential Anal., 26 (2007), 281-301. doi: 10.1007/s11118-006-9036-y.  Google Scholar

[2]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[3]

L. Bieberbach, $\Delta u= e^u$ und die automorphen Funktionen, Math. Ann., 77 (1916), 173-212. doi: 10.1007/BF01456901.  Google Scholar

[4]

O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: asymptotics, uniqueness and symmetry, J. Differential Equations, 249 (2010), 931-964. doi: 10.1016/j.jde.2010.02.023.  Google Scholar

[5]

M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations, 3 (1990), 1001-1014.  Google Scholar

[6]

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

[7]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125. doi: 10.1016/0362-546X(93)90012-H.  Google Scholar

[8]

W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc., 54 (1987), 141-175. doi: 10.1112/plms/s3-54.1.141.  Google Scholar

[9]

M. Feldman, Variational evolution problems and nonlocal geometric motion, Arch. Ration. Mech. Anal., 146 (1999), 221-274. doi: 10.1007/s002050050142.  Google Scholar

[10]

Y. Giga, "Surface Evolution Equations. A Level Set Approach," Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[11]

P. Juutinen, The boundary Harnack inequality for infinite harmonic functions in Lipschitz domains satisfying the interior ball condition, Nonlinear Anal., 69 (2008), 1941-1944. doi: 10.1016/j.na.2007.07.035.  Google Scholar

[12]

P. Juutinen and J. D. Rossi, Large solutions for the infinity Laplacian, Adv. Calc. Var., 1 (2008), 271-289. doi: 10.1515/ACV.2008.011.  Google Scholar

[13]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985.  Google Scholar

[14]

J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar

[15]

Y. Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math., 58 (2005), 85-146. doi: 10.1002/cpa.20051.  Google Scholar

[16]

R. Osserman, On the inequality $\Delta u\ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647. doi: 10.2140/pjm.1957.7.1641.  Google Scholar

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