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Borderline regularity for fully nonlinear equations in Dini domains

  • Karthik Adimurthi ORCID logo EMAIL logo and Agnid Banerjee
Published/Copyright: September 9, 2020
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 15 Issue 4

Abstract

In this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C 1 , Dini -domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention that, differently from the approach used there which is based on W 1 , q estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Luis Caffarelli.

Keywords: Fully nonlinear equations; borderline gradient continuity
MSC 2010: 35J25; 35J60

Funding source: Department of Atomic Energy, Government of India

Award Identifier / Grant number: 12-R & D-TFR-5.01-0520

Funding source: Science and Engineering Research Board

Award Identifier / Grant number: MTR/2018/000267

Funding statement: The first author was supported in part by the Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520. The second author is supported in part by SERB Matrix grant MTR/2018/000267 and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520.

Acknowledgements

We would like to thank the referee for his/her insightful comments which have contributed to improving the presentation of the paper.

  1. Communicated by: Luis Silvestre

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Received: 2020-04-05
Revised: 2020-06-28
Accepted: 2020-08-19
Published Online: 2020-09-09
Published in Print: 2022-10-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
  6. Integral representation for energies in linear elasticity with surface discontinuities
  7. Minkowski inequalities and constrained inverse curvature flows in warped spaces
  8. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
  9. Pairings between bounded divergence-measure vector fields and BV functions
  10. Equivalence between distributional and viscosity solutions for the double-phase equation
  11. Vanishing John–Nirenberg spaces
  12. Extremals in nonlinear potential theory
  13. On the structure of divergence-free measures on ℝ2
  14. Solutions of the (free boundary) Reifenberg Plateau problem
  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
https://doi.org/10.1515/acv-2020-0030
Keywords for this article
Fully nonlinear equations; borderline gradient continuity

Articles in the same Issue

  1. Frontmatter
  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
  6. Integral representation for energies in linear elasticity with surface discontinuities
  7. Minkowski inequalities and constrained inverse curvature flows in warped spaces
  8. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
  9. Pairings between bounded divergence-measure vector fields and BV functions
  10. Equivalence between distributional and viscosity solutions for the double-phase equation
  11. Vanishing John–Nirenberg spaces
  12. Extremals in nonlinear potential theory
  13. On the structure of divergence-free measures on ℝ2
  14. Solutions of the (free boundary) Reifenberg Plateau problem
  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
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