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

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.

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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

Mots-clés : 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

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     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://www.numdam.org/articles/10.1016/j.anihpc.2020.07.005/}
}

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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://www.numdam.org/articles/10.1016/j.anihpc.2020.07.005/

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