Regularity of the stress field for degenerate and/or singular elliptic problems

Benjamin Lledos

doi:10.5802/ambp.427

Regularity of the stress field for degenerate and/or singular elliptic problems
[Régularité du stress field pour des équations elliptiques dégénérées et/ou singulières]

Benjamin Lledos ¹

¹ Université de Toulouse Institut de Mathématiques de Toulouse CNRS UMR 5219 31062 Toulouse Cedex 9, France

Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 1, pp. 83-135.

Résumé
Abstract

Nous étudions la régularité des solutions d’équations elliptiques dégénérées et/ou singulières. Nous prouvons la continuité de $G (\nabla u)$ où $u$ est une solution localement Lipschitz de $div G (\nabla u) = λ \in ℝ$ en dimension deux sous certaines hypothèses de croissance sur $G$ . De plus, nous établissons un résultat valable en toute dimension, indiquant que la séparation entre $\nabla u$ et l’ensemble de dégénérescence de $G$ est continue.

We investigate the regularity of the solutions to degenerate and/or singular elliptic equations. We prove the continuity of $G (\nabla u)$ where $u$ is a locally Lipschitz solution of $div G (\nabla u) = λ \in ℝ$ in dimension two under some growth assumptions on $G$ . Additionally, we establish a result that holds in any dimension, indicating that the separation between $\nabla u$ and the degeneracy set of $G$ is continuous.

Publié le : 2024-10-23

DOI : 10.5802/ambp.427

Classification : 35B65, 35J62, 35J70, 35J75, 49N99
Mots clés : Regularity, elliptic PDE, dimension two

Affiliations des auteurs :

Benjamin Lledos ¹

¹ Université de Toulouse Institut de Mathématiques de Toulouse CNRS UMR 5219 31062 Toulouse Cedex 9, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AMBP_2024__31_1_83_0,
     author = {Benjamin Lledos},
     title = {Regularity of the stress field for degenerate and/or singular elliptic problems},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {83--135},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {31},
     number = {1},
     year = {2024},
     doi = {10.5802/ambp.427},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.427/}
}

TY  - JOUR
AU  - Benjamin Lledos
TI  - Regularity of the stress field for degenerate and/or singular elliptic problems
JO  - Annales mathématiques Blaise Pascal
PY  - 2024
SP  - 83
EP  - 135
VL  - 31
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.427/
DO  - 10.5802/ambp.427
LA  - en
ID  - AMBP_2024__31_1_83_0
ER  -

%0 Journal Article
%A Benjamin Lledos
%T Regularity of the stress field for degenerate and/or singular elliptic problems
%J Annales mathématiques Blaise Pascal
%D 2024
%P 83-135
%V 31
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.427/
%R 10.5802/ambp.427
%G en
%F AMBP_2024__31_1_83_0

Benjamin Lledos. Regularity of the stress field for degenerate and/or singular elliptic problems. Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 1, pp. 83-135. doi : 10.5802/ambp.427. https://ambp.centre-mersenne.org/articles/10.5802/ambp.427/

Bibliographie
Cité par

[1] Gabriele Anzellotti; Mariano Giaquinta Convex functionals and partial regularity, Arch. Ration. Mech. Anal., Volume 102 (1988) no. 3, pp. 243-272 | DOI | Zbl

[2] Benny Avelin; Tuomo Kuusi; Giuseppe Mingione Nonlinear Calderón-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 2, pp. 663-714 | DOI | Zbl

[3] John M. Ball A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988) (Lecture Notes in Physics), Volume 344, Springer, 1989, pp. 207-215 | DOI | Zbl

[4] Verena Bögelein; Frank Duzaar; Raffaella Giova; Antonia Passarelli di Napoli Higher regularity in congested traffic dynamics, Math. Ann., Volume 385 (2023) no. 3-4, pp. 1823-1878 | Zbl

[5] Pierre Bousquet Another look to the orthotropic functional in the plane, “Bruno Pini” Mathematical Analysis Seminar 2020, Università di Bologna, Alma Mater Studiorum, 2021, pp. 1-29 | Zbl

[6] Pierre Bousquet; Lorenzo Brasco $C^{1}$ regularity of orthotropic $p$ -harmonic functions in the plane, Anal. PDE, Volume 11 (2018) no. 4, pp. 813-854 | DOI | Zbl

[7] Lorenzo Brasco Global $L^{\infty}$ gradient estimates for solutions to a certain degenerate elliptic equation, Nonlinear Anal., Theory Methods Appl., Volume 74 (2011) no. 2, pp. 516-531 | DOI | Zbl

[8] Carsten Carstensen; Stefan Müller Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., Volume 34 (2002) no. 2, pp. 495-509 | DOI | Zbl

[9] Pietro Celada; Giovanni Cupini; Marcello Guidorzi Existence and regularity of minimizers of nonconvex integrals with $p - q$ growth, ESAIM, Control Optim. Calc. Var., Volume 13 (2007) no. 2, pp. 343-358 | DOI | Numdam | Zbl

[10] Arrigo Cellina Uniqueness and comparison results for functionals depending on $\nabla u$ and on $u$ , SIAM J. Optim., Volume 18 (2007) no. 3, pp. 711-716 | DOI

[11] Andrea Cianchi; Vladimir G. Maz’ya Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., Volume 229 (2018) no. 2, pp. 569-599 | DOI | Zbl

[12] Andrea Cianchi; Vladimir G. Maz’ya Optimal second-order regularity for the $p$ -Laplace system, J. Math. Pures Appl., Volume 132 (2019), pp. 41-78 | DOI | Zbl

[13] Francis Clarke Functional analysis, calculus of variations and optimal control, Graduate Texts in Mathematics, 264, Springer, 2013 | DOI

[14] Maria Colombo; Alessio Figalli Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., Volume 101 (2014) no. 1, pp. 94-117 | DOI | Zbl

[15] Daniela de Silva; Ovidiu Savin Minimizers of convex functionals arising in random surfaces, Duke Math. J., Volume 151 (2010) no. 3, pp. 487-532 | Zbl

[16] Luca Esposito; Giuseppe Mingione; Cristina Trombetti On the Lipschitz regularity for certain elliptic problems, Forum Math., Volume 18 (2006) no. 2, pp. 263-292 | Zbl

[17] Gerald B. Folland Introduction to partial differential equations, Princeton University Press, 1995, xii+324 pages

[18] Irene Fonseca; Nicola Fusco; Paolo Marcellini An existence result for a nonconvex variational problem via regularity, ESAIM, Control Optim. Calc. Var., Volume 7 (2002), pp. 69-95 | DOI | Numdam | Zbl

[19] David Gilbarg; Neil S. Trudinger Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, xiv+517 pages (Reprint of the 1998 edition) | DOI

[20] Enrico Giusti Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhäuser, 1984, xii+240 pages | DOI

[21] Umberto Guarnotta; Sunra Mosconi A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form, Anal. PDE, Volume 16 (2023) no. 8, pp. 1955-1988 | DOI | Zbl

[22] Philip Hartman; Louis Nirenberg On spherical image maps whose Jacobians do not change sign, Am. J. Math., Volume 81 (1959), pp. 901-920 | DOI | Zbl

[23] Bernhard Kawohl; Jana Stara; Gabriel Wittum Analysis and numerical studies of a problem of shape design, Arch. Ration. Mech. Anal., Volume 114 (1991) no. 4, pp. 349-363 | DOI | Zbl

[24] Andreas Kriegl; Peter W. Michor The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, 1997, x+618 pages | DOI

[25] Henri Lebesgue Sur le problème de dirichlet, Rend. Circ. Mat. Palermo, Volume 24 (1907), pp. 371-402 | DOI | Zbl

[26] Peter Lindqvist; Diego Ricciotti Regularity for an anisotropic equation in the plane, Nonlinear Anal., Theory Methods Appl., Volume 177 (2018), pp. 628-636 | DOI | Zbl

[27] Benjamin Lledos A uniqueness result for a translation invariant problem in the calculus of variations, J. Convex Anal., Volume 31 (2024) no. 1, pp. 121-130 | Zbl

[28] Connor Mooney Minimizers of convex functionals with small degeneracy set, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 2, 74, 19 pages | Zbl

[29] Diego Ricciotti Regularity of the derivatives of $p$ -orthotropic functions in the plane for $1 < p < 2$ , Ann. Acad. Sci. Fenn., Math., Volume 44 (2019) no. 2, pp. 1093-1099 | DOI | Zbl

[30] Filippo Santambrogio; Vincenzo Vespri Continuity in two dimensions for a very degenerate elliptic equation, Nonlinear Anal., Theory Methods Appl., Volume 73 (2010) no. 12, pp. 3832-3841 | DOI | Zbl

Cité par Sources :