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November  2018, 17(6): 2773-2788. doi: 10.3934/cpaa.2018131

## Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

 Mathematisches Institut der Universität zu Köln, Weyertal 86-90, Cologne, 50931, Germany

Received  October 2017 Revised  February 2018 Published  June 2018

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution
 $u$
of
 $-|\nabla u|^α Δ^N_p u = 1$
has the property that
 $u^\frac{α+1}{α+2}$
is a concave function. Secondly we consider positive solutions of the eigenvalue problem
 $-|\nabla u|^α Δ^N_p u = λ |u|^α u,$
in which case
 $\log u$
turns out to be concave. The methods provided include a weak comparison principle and a Hopf-type Lemma.
Citation: Michael Kühn. Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2773-2788. doi: 10.3934/cpaa.2018131
##### References:
 [1] O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, Journal de Math´ematiques Pures et Appliquées, 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7. Google Scholar [2] M. Bianchini and P. Salani, Power concavity for solutions of nonlinear elliptic problems in convex domains, In Geometric Properties for Parabolic and Elliptic PDE's, pages 35-48. Springer, 2013. doi: 10.1007/978-88-470-2841-8_3. Google Scholar [3] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389. Google Scholar [4] L. A. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Communications in Partial Differential Equations, 7 (1982), 1337-1379. doi: 10.1080/03605308208820254. Google Scholar [5] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.2307/1999343. Google Scholar [7] G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Archive for Rational Mechanics and Analysis, 218 (2015), 1577-1607. doi: 10.1007/s00205-015-0888-4. Google Scholar [8] G. Crasta and I. Fragalà, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proceedings of the American Mathematical Society, 144 (2016), 2547-2558. doi: 10.1090/proc/12916. Google Scholar [9] L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calculus of Variations and Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1. Google Scholar [10] B. Ishige and P. Salani, A note on parabolic power concavity, Kodai Mathematical Journal, 37 (2014), 668-679. doi: 10.2996/kmj/1414674615. Google Scholar [11] B. Kawohl, When are superharmonic functions concave? Applications to the St. Venant torsion problem and to the fundamental mode of the clamped membrane, Zeitschrift für Angewandte Mathematik und Mechanik, 64 (1984), T364-T366. Google Scholar [12] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, volume 1150 of Lecture Notes in Mathematics, Springer, 1985. doi: 10.1007/BFb0075060. Google Scholar [13] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Communications in Partial Differential Equations, 10 (1985), 1213-1225. doi: 10.1080/03605308508820404. Google Scholar [14] B. Kawohl and J. Horák, On the geometry of the p-Laplacian operator, Discrete and Continuous Dynamical Systems. Series S, 10 (2017), 799-813. doi: 10.3934/dcdss.2017040. Google Scholar [15] A. Kennington, Power concavity and boundary value problems, Indiana U. Math. J, 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036. Google Scholar [16] N. Korevaar, Capillary surface convexity above convex domains, Indiana University Mathematics Journal, 32 (1983), 73-81. doi: 10.1512/iumj.1983.32.32007. Google Scholar [17] N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana University Mathematics Journal, 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042. Google Scholar [18] M. Kühn, On Viscosity Solutions and the Normalized p-Laplacian, PhD thesis, University of Cologne, 2017.Google Scholar [19] T. Kulczycki, On concavity of solutions of the dirichlet problem for the equation ${\left( { - \Delta u} \right)^{\frac{1}{2}}}\varphi = 1$ in convex planar regions, Journal of the European Mathematical Society, 19 (2017), 1361-1420. doi: 10.4171/JEMS/695. Google Scholar [20] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Communications in Partial Differential Equations, 33 (2008), 1788-1817. doi: 10.1080/03605300802289253. Google Scholar [21] G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, In Lecture Notes of Seminario Interdisciplinare di Matematica 7, Conference on Geometric Methods in PDE's, On the Occasion of 65th Birthday of Ermanno Lanconelli (Bologna), pages 207-222, Potenza: Università degli Studi della Basilicata, Dipartimento die Matematica e Informatica, 2008. Google Scholar [22] L. G. Makar-Limanov, Solution of Dirichlet's problem for the equation Δu = −1 in a convex region, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 52-53. Google Scholar [23] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 14 (1987), 403-421. Google Scholar [24] L. Zhao, Power concavity for doubly nonlinear parabolic equations, Journal of Mathematical Study. Shuxue Yanjiu, 50 (2017), 190-198. doi: 10.4208/jms.v50n2.17.05. Google Scholar

show all references

##### References:
 [1] O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, Journal de Math´ematiques Pures et Appliquées, 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7. Google Scholar [2] M. Bianchini and P. Salani, Power concavity for solutions of nonlinear elliptic problems in convex domains, In Geometric Properties for Parabolic and Elliptic PDE's, pages 35-48. Springer, 2013. doi: 10.1007/978-88-470-2841-8_3. Google Scholar [3] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389. Google Scholar [4] L. A. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Communications in Partial Differential Equations, 7 (1982), 1337-1379. doi: 10.1080/03605308208820254. Google Scholar [5] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.2307/1999343. Google Scholar [7] G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Archive for Rational Mechanics and Analysis, 218 (2015), 1577-1607. doi: 10.1007/s00205-015-0888-4. Google Scholar [8] G. Crasta and I. Fragalà, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proceedings of the American Mathematical Society, 144 (2016), 2547-2558. doi: 10.1090/proc/12916. Google Scholar [9] L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calculus of Variations and Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1. Google Scholar [10] B. Ishige and P. Salani, A note on parabolic power concavity, Kodai Mathematical Journal, 37 (2014), 668-679. doi: 10.2996/kmj/1414674615. Google Scholar [11] B. Kawohl, When are superharmonic functions concave? Applications to the St. Venant torsion problem and to the fundamental mode of the clamped membrane, Zeitschrift für Angewandte Mathematik und Mechanik, 64 (1984), T364-T366. Google Scholar [12] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, volume 1150 of Lecture Notes in Mathematics, Springer, 1985. doi: 10.1007/BFb0075060. Google Scholar [13] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Communications in Partial Differential Equations, 10 (1985), 1213-1225. doi: 10.1080/03605308508820404. Google Scholar [14] B. Kawohl and J. Horák, On the geometry of the p-Laplacian operator, Discrete and Continuous Dynamical Systems. Series S, 10 (2017), 799-813. doi: 10.3934/dcdss.2017040. Google Scholar [15] A. Kennington, Power concavity and boundary value problems, Indiana U. Math. J, 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036. Google Scholar [16] N. Korevaar, Capillary surface convexity above convex domains, Indiana University Mathematics Journal, 32 (1983), 73-81. doi: 10.1512/iumj.1983.32.32007. Google Scholar [17] N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana University Mathematics Journal, 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042. Google Scholar [18] M. Kühn, On Viscosity Solutions and the Normalized p-Laplacian, PhD thesis, University of Cologne, 2017.Google Scholar [19] T. Kulczycki, On concavity of solutions of the dirichlet problem for the equation ${\left( { - \Delta u} \right)^{\frac{1}{2}}}\varphi = 1$ in convex planar regions, Journal of the European Mathematical Society, 19 (2017), 1361-1420. doi: 10.4171/JEMS/695. Google Scholar [20] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Communications in Partial Differential Equations, 33 (2008), 1788-1817. doi: 10.1080/03605300802289253. Google Scholar [21] G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, In Lecture Notes of Seminario Interdisciplinare di Matematica 7, Conference on Geometric Methods in PDE's, On the Occasion of 65th Birthday of Ermanno Lanconelli (Bologna), pages 207-222, Potenza: Università degli Studi della Basilicata, Dipartimento die Matematica e Informatica, 2008. Google Scholar [22] L. G. Makar-Limanov, Solution of Dirichlet's problem for the equation Δu = −1 in a convex region, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 52-53. Google Scholar [23] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 14 (1987), 403-421. Google Scholar [24] L. Zhao, Power concavity for doubly nonlinear parabolic equations, Journal of Mathematical Study. Shuxue Yanjiu, 50 (2017), 190-198. doi: 10.4208/jms.v50n2.17.05. Google Scholar
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