We obtain improved fractional differentiability of solutions to the Banach-space valued Finsler -Laplacian defined on a -convex, -smooth Banach space. The operators we consider are non-linear and very degenerately elliptic. Our results are new already in the -valued setting.
Revised:
Accepted:
Published online:
Max Goering 1; Lukas Koch 1
CC-BY 4.0
@article{CRMATH_2023__361_G7_1091_0,
author = {Max Goering and Lukas Koch},
title = {A note on improved differentiability for the {Banach-space} valued {Finsler} $\gamma ${-Laplacian}},
journal = {Comptes Rendus. Math\'ematique},
pages = {1091--1105},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
year = {2023},
doi = {10.5802/crmath.474},
language = {en},
}
TY - JOUR AU - Max Goering AU - Lukas Koch TI - A note on improved differentiability for the Banach-space valued Finsler $\gamma $-Laplacian JO - Comptes Rendus. Mathématique PY - 2023 SP - 1091 EP - 1105 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.474 LA - en ID - CRMATH_2023__361_G7_1091_0 ER -
Max Goering; Lukas Koch. A note on improved differentiability for the Banach-space valued Finsler $\gamma $-Laplacian. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1091-1105. doi : 10.5802/crmath.474. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.474/
[1] Nonlinear Calderón-Zygmund Theory in the Limiting Case, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 2, pp. 663-714 | DOI | Zbl
[2] Higher Order Regularity Shifts for the -Poisson Problem, Numerical Mathematics and Advanced Applications ENUMATH 2019 (Lecture Notes in Computational Science and Engineering), Volume 139, Springer, 2021, pp. 1147-1155 | DOI | MR | Zbl
[3] On the rate of convergence of iterated bregman projections and of the alternating algorithm, J. Math. Anal. Appl., Volume 481 (2020) no. 1, 123482, 23 pages | MR | Zbl
[4] Isoperimetric inequalities, wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., Volume 54 (2003) no. 5, pp. 771-783 | DOI | MR | Zbl
[5] p-harmonic equation and quasiregular mappings, Partial differential equations (Banach Center Publications), Volume 19, PWN - Polish Scientific Publishers, 1987, pp. 25-38 | MR | Zbl
[6] The nonoccurrence of the Lavrentiev phenomenon for a class of variational functionals, SIAM J. Control Optim., Volume 51 (2013) no. 2, pp. 1639-1650 | DOI | MR | Zbl
[7] The higher integrability and the validity of the Euler-Lagrange equation for solutions to variational problems, SIAM J. Control Optim., Volume 50 (2012) no. 2, pp. 888-899 | DOI | MR | Zbl
[8] Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., Volume 93 (2010) no. 6, pp. 652-671 | DOI
[9] Sobolev and lipschitz regularity for local minimizers of widely degenerate anisotropic functionals, Nonlinear Anal., Theory Methods Appl., Volume 153 (2017), pp. 169-199 | DOI | MR | Zbl
[10] Regularity of minimisers of autonomous convex variational integrals, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 13 (2014) no. 4, pp. 1065-1089 | Zbl
[11] Regularity results via duality for minimizers of degenerate functionals, Asymptotic Anal., Volume 44 (2005) no. 3-4, pp. 221-235 | MR | Zbl
[12] Regularity for minimizers of degenerate elliptic functionals, J. Nonlinear Convex Anal., Volume 7 (2006) no. 3, pp. 375-383 | MR | Zbl
[13] Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems, Arch. Ration. Mech. Anal., Volume 229 (2018) no. 2, pp. 569-599 | DOI | MR | Zbl
[14] Uniformly convex spaces, Trans. Am. Math. Soc., Volume 40 (1936) no. 3, pp. 396-414 | DOI | MR | Zbl
[15] Regularity results for very degenerate elliptic equations, Flows of Non-smooth Vector Fields and Degenerate Elliptic Equations, Springer, 2017, pp. 119-157 | DOI
[16] Bubbling with -almost constant mean curvature and an alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal., Volume 230 (2018) no. 3, pp. 1131-1177 | DOI | MR | Zbl
[17] Lipschitz interior regularity for the viscosity and weak solutions of the pseudo -laplacian equation, Adv. Differ. Equ., Volume 21 (2016) no. 3-4, pp. 373-400 | MR | Zbl
[18] Eigenvalues of the finsler p-laplacian on varying domains, Mathematika, Volume 66 (2020) no. 3, pp. 765-776 | DOI | MR | Zbl
[19] Remarks on a Finsler–Laplacian, Proc. Am. Math. Soc., Volume 137 (2009) no. 1, pp. 247-253 | DOI | MR | Zbl
[20] Regularity in relaxed convex problems (in preparation)
[21] Direct Methods in the Calculus of Variations, World Scientific, 2003 | DOI | Zbl
[22] A harnack inequality for -solutions of anisotropic pdes with non-standard growth (2020) | arXiv
[23] A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form (2021) | arXiv
[24] Analysis in Banach spaces. Volume I: Martingales and Littlewood-Paley theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 63, Springer, 2016 | DOI
[25] The -laplace eigenvalue problem as and Cheeger sets in a Finsler metric, J. Convex Anal., Volume 15 (2008) no. 3, pp. 623-634 | MR | Zbl
[26] On the validity of the Euler-Lagrange system without growth assumptions (2022) | arXiv
[27] Sobolev spaces of vector-valued functions, Ph. D. Thesis, University of Ulm (2015)
[28] Nonlinear spectrums of finsler manifolds, Math. Z., Volume 300 (2022) no. 1, pp. 81-123 | DOI | MR | Zbl
[29] Classical Banach spaces II: function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer, 2013
[30] Differential properties of solutions of variational problems for functionals of linear growth, J. Sov. Math., Volume 64 (1993) no. 6, pp. 1256-1277 | DOI
[31] Sharp spectral gap for the Finsler -Laplacian, Sci. China, Math., Volume 62 (2019) no. 8, pp. 1615-1644 | MR | Zbl
[32] Characteristic inequalities of uniformly convex and uniformly smooth banach spaces, J. Math. Anal. Appl., Volume 157 (1991) no. 1, pp. 189-210 | MR | Zbl
[33] Homogenization of differential operators and integral functionals, Springer, 1994
Cited by Sources:
Comments - Policy