Symmetry and monotonicity results for solutions of vectorial 𝑝-Stokes systems

López-Soriano, Rafael; Montoro, Luigi; Sciunzi, Berardino

doi:10.1090/tran/8867

Symmetry and monotonicity results for solutions of vectorial $p$-Stokes systems
HTML articles powered by AMS MathViewer

by Rafael López-Soriano, Luigi Montoro and Berardino Sciunzi;

Trans. Amer. Math. Soc. 376 (2023), 3493-3514

DOI: https://doi.org/10.1090/tran/8867

Published electronically: February 28, 2023

HTML | PDF | Request permission

Abstract:

In this paper we shall study qualitative properties of a $p$-Stokes type system, namely \begin{equation*} -{\boldsymbol \Delta }_p{\boldsymbol u}=-\operatorname {\mathbf {div}}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u}) \text { in $\Omega $}, \end{equation*} where ${\boldsymbol \Delta }_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $\Omega$ and the function $\boldsymbol { f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators.

References

Anna Kh. Balci, Andrea Cianchi, Lars Diening, and Vladimir Maz’ya, A pointwise differential inequality and second-order regularity for nonlinear elliptic systems, Math. Ann. 383 (2022), no. 3-4, 1775–1824. MR 4458389, DOI 10.1007/s00208-021-02249-9
H. Beirão da Veiga and F. Crispo, On the global $W^{2,q}$ regularity for nonlinear $N$-systems of the $p$-Laplacian type in $n$ space variables, Nonlinear Anal. 75 (2012), no. 11, 4346–4354. MR 2921994, DOI 10.1016/j.na.2012.03.021
Hugo Beirão da Veiga and Francesca Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 5, 1173–1191. MR 3039691, DOI 10.3934/dcdss.2013.6.1173
Luigi C. Berselli, Lars Diening, and Michael Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech. 12 (2010), no. 1, 101–132. MR 2602916, DOI 10.1007/s00021-008-0277-y
R. Byron Bird, Polymeric liquids: from molecular models to constitutive equations, Viscoelasticity and rheology (Madison, Wis., 1984) Academic Press, Orlando, FL, 1985, pp. 105–123. MR 830199
Andrea Cianchi and Vladimir G. Maz’ya, Optimal second-order regularity for the $p$-Laplace system, J. Math. Pures Appl. (9) 132 (2019), 41–78 (English, with English and French summaries). MR 4030249, DOI 10.1016/j.matpur.2019.02.015
F. Crispo and P. Maremonti, A high regularity result of solutions to modified $p$-Stokes equations, Nonlinear Anal. 118 (2015), 97–129. MR 3325608, DOI 10.1016/j.na.2014.10.017
Lucio Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 493–516 (English, with English and French summaries). MR 1632933, DOI 10.1016/S0294-1449(98)80032-2
Lucio Damascelli and Filomena Pacella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1<p<2$, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, 689–707. MR 1648566
Lucio Damascelli and Berardino Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations 206 (2004), no. 2, 483–515. MR 2096703, DOI 10.1016/j.jde.2004.05.012
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
Jacques-Louis Lions, Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France 93 (1965), 155–175 (French). MR 194760, DOI 10.24033/bsmf.1620
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 259693
Tuomo Kuusi and Giuseppe Mingione, A nonlinear Stein theorem, Calc. Var. Partial Differential Equations 51 (2014), no. 1-2, 45–86. MR 3247381, DOI 10.1007/s00526-013-0666-9
Tuomo Kuusi and Giuseppe Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 929–1004. MR 3779689, DOI 10.4171/JEMS/780
J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Evolutionary equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, pp. 371–459. MR 2182831
J. Málek, K. R. Rajagopal, and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci. 5 (1995), no. 6, 789–812. MR 1348587, DOI 10.1142/S0218202595000449
Giuseppe Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), no. 4, 355–426. MR 2291779, DOI 10.1007/s10778-006-0110-3
Patrizia Pucci and James Serrin, The maximum principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007. MR 2356201, DOI 10.1007/978-3-7643-8145-5
Giovanni Maria Troianiello, Elliptic differential equations and obstacle problems, The University Series in Mathematics, Plenum Press, New York, 1987. MR 1094820, DOI 10.1007/978-1-4899-3614-1