Steiner symmetrization for anisotropic quasilinear equations via partial discretization

Brock, F.; Díaz, J.I.; Ferone, A.; Gómez-Castro, D.; Mercaldo, A.

doi:10.1016/j.anihpc.2020.07.005

Geodesic

Parcourir par

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

Brock, F.¹ ; Díaz, J.I.^2,³ ; Ferone, A.⁴ ; Gómez-Castro, D.^2,³ ; Mercaldo, A.⁵

¹ a Institute of Mathematics, University of Rostock, Germany
² b Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Spain
³ c Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain
⁴ d Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”, Italy
⁵ e Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Italy

Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 347-368.

Voir la notice de l'article provenant de la source Numdam

Résumé

In this paper we obtain comparison results for the quasilinear equation $- Δ_{p, x} u - u_{y y} = f$ with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable x, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in y, and the proof of a comparison principle for the discrete version of the auxiliary problem $A U - U_{y y} \leq \int_{0}^{s} f$ , where $A U = {(n ω_{n}^{1 / n} s^{1 / n^{'}})}^{p} {(- U_{s s})}^{p - 1}$ . We show that this operator is T-accretive in $L^{\infty}$ . We extend our results for $- Δ_{p, x}$ to general operators of the form $- div (a (| \nabla_{x} u |) \nabla_{x} u)$ where a is non-decreasing and behaves like $| \cdot |^{p - 2}$ at infinity.

DOI : 10.1016/j.anihpc.2020.07.005

Keywords: Steiner symmetrization, Anisotropic quasilinear equations, Partial discretization, T-accretive operators

Affiliations des auteurs :

Brock, F. ¹ ; Díaz, J.I. ^{2, 3} ; Ferone, A. ⁴ ; Gómez-Castro, D. ^{2, 3} ; Mercaldo, A. ⁵

@article{AIHPC_2021__38_2_347_0,
     author = {Brock, F. and D{\'\i}az, J.I. and Ferone, A. and G\'omez-Castro, D. and Mercaldo, A.},
     title = {Steiner symmetrization for anisotropic quasilinear equations via partial discretization},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {347--368},
     publisher = {Elsevier},
     volume = {38},
     number = {2},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.07.005},
     mrnumber = {4211989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.005/}
}

TY  - JOUR
AU  - Brock, F.
AU  - Díaz, J.I.
AU  - Ferone, A.
AU  - Gómez-Castro, D.
AU  - Mercaldo, A.
TI  - Steiner symmetrization for anisotropic quasilinear equations via partial discretization
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 347
EP  - 368
VL  - 38
IS  - 2
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.005/
DO  - 10.1016/j.anihpc.2020.07.005
LA  - en
ID  - AIHPC_2021__38_2_347_0
ER  -

%0 Journal Article
%A Brock, F.
%A Díaz, J.I.
%A Ferone, A.
%A Gómez-Castro, D.
%A Mercaldo, A.
%T Steiner symmetrization for anisotropic quasilinear equations via partial discretization
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 347-368
%V 38
%N 2
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.005/
%R 10.1016/j.anihpc.2020.07.005
%G en
%F AIHPC_2021__38_2_347_0

Brock, F.; Díaz, J.I.; Ferone, A.; Gómez-Castro, D.; Mercaldo, A. Steiner symmetrization for anisotropic quasilinear equations via partial discretization. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 347-368. doi : 10.1016/j.anihpc.2020.07.005. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.005/

Bibliographie
Cité par

[1] Alvino, A.; Díaz, J.I.; Lions, P.-L.; Trombetti, G. Équations elliptiques et symètrisation de Steiner, C. R. Acad. Sci. Paris, Volume 314 (1992), pp. 1015-1020 | MR | Zbl

[2] Alvino, A.; Ferone, V.; Trombetti, G.; Lions, P.-L. Convex symmetrization and applications, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 14 (1997) no. 2, pp. 275-293 | MR | Zbl | mathdoc-id | DOI

[3] Alvino, A.; Trombetti, G. A class of degenerate nonlinear elliptic equations, Ric. Mat., Volume 29 (1980) no. 2, pp. 193-212 | MR | Zbl

[4] Alvino, A.; Trombetti, G.; Díaz, J.I.; Lions, P.L. Elliptic equations and Steiner symmetrization, Commun. Pure Appl. Math., Volume 49 (1996) no. 3, pp. 217-236 | MR | Zbl | 3.0.CO;2-G class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[5] Alvino, A.; Trombetti, G.; Lions, P.L. Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 7 (1990) no. 2, pp. 37-65 | MR | Zbl | mathdoc-id | DOI

[6] Antontsev, S.N.; Díaz, J.I.; Shmarev, S. Energy Methods for Free Boundary Problems, Birkhäuser, Boston, 2002 | MR | Zbl | DOI

[7] Baernstein, A.; Taylor, B.A. Spherical rearrangements, subharmonic functions, and *-functions in $n$ -space, Duke Math. J., Volume 43 (1976) no. 2, pp. 245-268 | MR | Zbl | DOI

[8] C. Bandle, B. Kawohl, Application de la symétrization de steiner aux problèmes de poisson, preprint, 1992.

[9] Barbu, V. Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer New York, New York, NY, 2010 | MR | Zbl | DOI

[10] Benilan, P.; Spiteri, P. Discretisation par schema implicite d'un probleme abstrait $d^{2} u / d t^{2} \in A u$ sur $[0, 1]$ , $u (0) = u_{0}, u (1) = u_{1}$ , Publ. Math. Besanson Anal. Nonlin., Volume 10 (1987), pp. 93-100 | Zbl

[11] Brezis, H. Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, Elsevier, 1971, pp. 101-156 | MR | Zbl | DOI

[12] Brock, F. Continuous Steiner-symmetrization, Math. Nachr., Volume 172 (1995), pp. 25-48 | MR | Zbl | DOI

[13] Brock, F. Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci., Volume 110 (2000) no. 2, pp. 157-204 | MR | Zbl | DOI

[14] Brock, F.; Chiacchio, F.; Ferone, A.; Mercaldo, A. New Pólya–Szegö-type inequalities and an alternative approach to comparison results for PDE's, Adv. Math., Volume 336 (2018), pp. 316-334 | MR | DOI

[15] Burchard, A. Steiner symmetrization is continuous in $W^{1, p}$ , Geom. Funct. Anal., Volume 7 (1997) no. 5, pp. 823-860 | MR | Zbl | DOI

[16] Burchard, A.; Ferone, A. On the extremals of the Pólya-Szego inequality, Indiana Univ. Math. J., Volume 64 (2015) no. 5, pp. 1447-1463 | MR | DOI

[17] Chiacchio, F. Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization, Publ. Mat., Volume 53 (2009) no. 1, pp. 47-71 | MR | Zbl | DOI

[18] Cianchi, A.; Maz'ya, V.G. Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal., Volume 212 (2014) no. 1, pp. 129-177 | MR | Zbl | DOI

[19] Cianchi, A.; Maz'ya, V.G. Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., Volume 229 (2018) no. 2, pp. 569-599 | MR | DOI

[20] Crandall, M.G.; Ishii, H.; Lions, P.-L. User's guide to viscosity solutions of second order partial differential equation, Bull. Am. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67 | MR | Zbl | DOI

[21] Dal Maso, G. An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations, Birkhäuser Boston, Boston, MA, 1993 | MR | Zbl | DOI

[22] Díaz, J.I. Simetrización de problemas parabólicos no lineales: aplicación a ecuaciones de reacción-difusión, Madrid (Memorias de la Real Academia de Ciencias Exactas, Físicas y Naturales), Volume vol. XXVIII (1991) | Zbl

[23] Díaz, J.I.; Gómez-Castro, D. On the effectiveness of wastewater cylindrical reactors: an analysis through Steiner symmetrization, Pure Appl. Geophys., Volume 173 (2016) no. 3

[24] Ekeland, I.; Temam, R. Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics, jan 1999 | MR | Zbl | DOI

[25] Evans, L.C. On solving certain nonlinear partial differential equations by accretive operator methods, Isr. J. Math., Volume 36 (1980) no. 3–4, pp. 225-247 | MR | Zbl | DOI

[26] Evans, L.C. Weak Convergence Methods for Nonlinear Partial Differential Equations. Number 74, American Mathematical Society, Providence, Rhode Island, 1988 | MR | Zbl

[27] Ferone, V.; Mercaldo, A. Neumann problems and Steiner symmetrization, Commun. Partial Differ. Equ., Volume 30 (2005) no. 10–12, pp. 1537-1553 | MR | Zbl | DOI

[28] Fonseca, I. The Wulff theorem revisited, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., Volume 432 (1991) no. 1884, pp. 125-145 | MR | Zbl

[29] Gehring, F.W. Symmetrization of rings in space, Trans. Am. Math. Soc., Volume 101 (1961) no. 3, p. 499 | MR | Zbl | DOI

[30] Ishii, H. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekvacioj, Volume 38 (1995), pp. 101-120 | MR | Zbl

[31] Juutinen, P.; Lindqvist, P.; Manfredi, J.J. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., Volume 33 (2003) no. 3, pp. 699-717 | MR | Zbl | DOI

[32] Lions, P.L.; Souganidis, P.E. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 5, pp. 667-677 | MR | Zbl | mathdoc-id | DOI

[33] Maz'ja, V.G. Weak solutions of the Dirichlet and Neumann problems, Tr. Mosk. Mat. Obš., Volume 20 (1969), pp. 137-172 | Zbl

[34] Medina, M.; Ochoa, P. On viscosity and weak solutions for non-homogeneous $p$ -Laplace equations, Adv. Nonlinear Anal., Volume 8 (2019) no. 1, pp. 468-481 | MR | DOI

[35] Meier, M. Boundedness and integrability properties of weak solutions of quasilinear elliptic systems, J. Reine Angew. Math., Volume 1982 (1982) no. 333, pp. 191-220 | MR | Zbl | DOI

[36] Minty, G.J. Monotone (nonlinear) operators in Hilbert space, Duke Math. J., Volume 29 (1962) no. 3, pp. 341-346 | MR | Zbl | DOI

[37] Newnham, R.E. Properties of Materials: Anisotropy, Symmetry, Structure, Oxford University Press, Oxford, 2005

[38] Poffald, E.I.; Reich, S. A quasi-autonomous second-order differential inclusion, Trends in Theory and Practice of Non-Linear Analysis, 1985, pp. 387-392 | MR | Zbl

[39] Reich, S.; Shafrir, I. An existence theorem for a difference inclusion in general Banach spaces, J. Math. Anal. Appl., Volume 160 (1991) no. 2, pp. 406-412 | MR | Zbl | DOI

[40] Talenti, G. Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 3 (1976) no. 4, pp. 697-718 | MR | Zbl | mathdoc-id

[41] Talenti, G. Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Appl. Anal., Volume 6 (1977) no. 4, pp. 319-320 | Zbl | DOI

[42] Talenti, G. Linear elliptic p.d.e.'s: level sets, rearrangements and a priori estimates of solutions, Boll. Unione Mat. Ital., B (6), Volume 4 (1985) no. 3, pp. 917-949 | MR | Zbl

[43] Talenti, G. The art of rearranging, Milan J. Math., Volume 84 (2016) no. 1, pp. 105-157 | MR | DOI

[44] Van Schaftingen, J. Anisotropic symmetrization, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 4, pp. 539-565 | MR | Zbl | mathdoc-id | DOI

[45] Weinberger, H.F. Symmetrization in uniformly elliptic problems, Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif, 1962, pp. 424-428 | MR | Zbl

[46] Wulff, G. Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Krystallflächen, Z. Kristallogr., Volume 34 (1901)

[47] Zhikov, V.V. On the homogenization technique for variational problems, Funct. Anal. Appl., Volume 33 (1999) no. 1, pp. 11-24 | Zbl | DOI

Cité par Sources :