We investigate the regularity of the solutions to degenerate and/or singular elliptic equations. We prove the continuity of where is a locally Lipschitz solution of in dimension two under some growth assumptions on . Additionally, we establish a result that holds in any dimension, indicating that the separation between and the degeneracy set of is continuous.
Nous étudions la régularité des solutions d’équations elliptiques dégénérées et/ou singulières. Nous prouvons la continuité de où est une solution localement Lipschitz de en dimension deux sous certaines hypothèses de croissance sur . De plus, nous établissons un résultat valable en toute dimension, indiquant que la séparation entre et l’ensemble de dégénérescence de est continue.
Keywords: Regularity, elliptic PDE, dimension two
Benjamin Lledos 1
@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, Volume 31 (2024) no. 1, pp. 83-135. doi : 10.5802/ambp.427. https://ambp.centre-mersenne.org/articles/10.5802/ambp.427/
[1] Convex functionals and partial regularity, Arch. Ration. Mech. Anal., Volume 102 (1988) no. 3, pp. 243-272 | DOI | Zbl
[2] Nonlinear Calderón-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 2, pp. 663-714 | DOI | Zbl
[3] 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] Higher regularity in congested traffic dynamics, Math. Ann., Volume 385 (2023) no. 3-4, pp. 1823-1878 | Zbl
[5] 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] regularity of orthotropic -harmonic functions in the plane, Anal. PDE, Volume 11 (2018) no. 4, pp. 813-854 | DOI | Zbl
[7] Global 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] Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., Volume 34 (2002) no. 2, pp. 495-509 | DOI | Zbl
[9] Existence and regularity of minimizers of nonconvex integrals with growth, ESAIM, Control Optim. Calc. Var., Volume 13 (2007) no. 2, pp. 343-358 | DOI | Numdam | Zbl
[10] Uniqueness and comparison results for functionals depending on and on , SIAM J. Optim., Volume 18 (2007) no. 3, pp. 711-716 | DOI
[11] Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., Volume 229 (2018) no. 2, pp. 569-599 | DOI | Zbl
[12] Optimal second-order regularity for the -Laplace system, J. Math. Pures Appl., Volume 132 (2019), pp. 41-78 | DOI | Zbl
[13] Functional analysis, calculus of variations and optimal control, Graduate Texts in Mathematics, 264, Springer, 2013 | DOI
[14] Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., Volume 101 (2014) no. 1, pp. 94-117 | DOI | Zbl
[15] Minimizers of convex functionals arising in random surfaces, Duke Math. J., Volume 151 (2010) no. 3, pp. 487-532 | Zbl
[16] On the Lipschitz regularity for certain elliptic problems, Forum Math., Volume 18 (2006) no. 2, pp. 263-292 | Zbl
[17] Introduction to partial differential equations, Princeton University Press, 1995, xii+324 pages
[18] 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] Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, xiv+517 pages (Reprint of the 1998 edition) | DOI
[20] Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhäuser, 1984, xii+240 pages | DOI
[21] 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] On spherical image maps whose Jacobians do not change sign, Am. J. Math., Volume 81 (1959), pp. 901-920 | DOI | Zbl
[23] 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] The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, 1997, x+618 pages | DOI
[25] Sur le problème de dirichlet, Rend. Circ. Mat. Palermo, Volume 24 (1907), pp. 371-402 | DOI | Zbl
[26] Regularity for an anisotropic equation in the plane, Nonlinear Anal., Theory Methods Appl., Volume 177 (2018), pp. 628-636 | DOI | Zbl
[27] 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] Minimizers of convex functionals with small degeneracy set, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 2, 74, 19 pages | Zbl
[29] Regularity of the derivatives of -orthotropic functions in the plane for , Ann. Acad. Sci. Fenn., Math., Volume 44 (2019) no. 2, pp. 1093-1099 | DOI | Zbl
[30] 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
Cited by Sources: