# American Institute of Mathematical Sciences

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

## Convexity of solutions to boundary blow-up problems

 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.
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.  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.  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.  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.  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.   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.  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.  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.  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.  doi: 10.1007/s002050050142.  Google Scholar [10] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Monographs in Mathematics, (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.  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.  doi: 10.1515/ACV.2008.011.  Google Scholar [13] B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, (1150).   Google Scholar [14] J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.  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.  doi: 10.1002/cpa.20051.  Google Scholar [16] R. Osserman, On the inequality $\Delta u\ge f(u)$,, Pacific J. Math., 7 (1957), 1641.  doi: 10.2140/pjm.1957.7.1641.  Google Scholar

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##### 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.  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.  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.  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.  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.   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.  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.  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.  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.  doi: 10.1007/s002050050142.  Google Scholar [10] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Monographs in Mathematics, (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.  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.  doi: 10.1515/ACV.2008.011.  Google Scholar [13] B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, (1150).   Google Scholar [14] J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.  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.  doi: 10.1002/cpa.20051.  Google Scholar [16] R. Osserman, On the inequality $\Delta u\ge f(u)$,, Pacific J. Math., 7 (1957), 1641.  doi: 10.2140/pjm.1957.7.1641.  Google Scholar
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