This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.
Citation: Antonio Greco, Francesco Pisanu. Improvements on overdetermined problems associated to the $ p $-Laplacian[J]. Mathematics in Engineering, 2022, 4(3): 1-14. doi: 10.3934/mine.2022017
This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.
| [1] |
M. Belloni, B. Kawohl, A direct uniqueness proof for equations involving the $p$-Laplace operator, Manuscripta Math., 109 (2002), 229–231. doi: 10.1007/s00229-002-0305-9
|
| [2] | M. Cuesta, P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differ. Integral Equ., 13 (2000), 721–746. |
| [3] | L. Damascelli, B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of $m$-Laplace equations, Calc. Var., 25 (2005), 139–159. |
| [4] | J. I. Díaz, J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521–524. |
| [5] | D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2 Eds., Springer-Verlag, 1998. |
| [6] |
A. Greco, Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian, Publ. Mat., 58 (2014), 485–498. doi: 10.5565/PUBLMAT_58214_24
|
| [7] |
A. Greco, Constrained radial symmetry for monotone elliptic quasilinear operators, J. Anal. Math., 121 (2013), 223–234. doi: 10.1007/s11854-013-0033-y
|
| [8] | A. Greco, Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387–399. |
| [9] |
A. Greco, V. Mascia, Non-local sublinear problems: existence, comparison, and radial symmetry, Discrete Contin. Dyn. Syst., 39 (2019), 503–519. doi: 10.3934/dcds.2019021
|
| [10] | J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, New York: The Clarendon Press, Oxford University Press, 1993. |
| [11] | P. Lindqvist, Notes on the stationary $p$-Laplace equation, Springer, 2019. |
| [12] |
M. Lucia, S. Prashanth, Strong comparison principle for solutions of quasilinear equations, P. Am. Math. Soc., 132 (2003), 1005–1011. doi: 10.1090/S0002-9939-03-07285-X
|
| [13] | M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, 1984. |
| [14] | P. Pucci, J. Serrin, The maximum principle, Birkhäuser, 2007. |
| [15] |
P. Pucci, B. Sciunzi, J. Serrin, Partial and full symmetry of solutions of quasilinear elliptic equations, via the Comparison Principle, Contemp. Math., 446 (2007), 437–444. doi: 10.1090/conm/446/08643
|
| [16] |
P. Roselli, B. Sciunzi, A strong comparison principle for the p-Laplacian, P. Am. Math. Soc., 135 (2007), 3217–3224. doi: 10.1090/S0002-9939-07-08847-8
|
| [17] |
B. Sciunzi, Regularity and comparison principles for $p$-Laplace equations with vanishing source term, Commun. Contemp. Math., 16 (2014), 1450013. doi: 10.1142/S0219199714500138
|
Antonio Greco, Francesco Pisanu. Improvements on overdetermined problems associated to the $ p $-Laplacian[J]. Mathematics in Engineering, 2022, 4(3): 1-14. doi: 10.3934/mine.2022017
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