Lipschitz regularity results for a class of obstacle problems with nearly linear growth

Abstract

This paper deals with the Lipschitz continuity of solutions to variational obstacle problems with nearly linear growth. The main tool used here is a new higher differentiability result which reveals to be crucial because it allows us to perform the linearization procedure to transform the constrained problem in an unconstrained one and it permits us to deduce the equivalence between our minimization problem and its corresponding variational formulation. Our results hold true for a large class of example for which the Lavrentiev phenomenon does not occur, not necessarily for lagrangians dependent on the modulus of the gradient. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.

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Correspondence to Samuele Riccò.

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Bertazzoni, G., Riccò, S. Lipschitz regularity results for a class of obstacle problems with nearly linear growth. J Elliptic Parabol Equ 6, 883–918 (2020). https://doi.org/10.1007/s41808-020-00088-4

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Keywords

  • Regularity of solutions
  • Obstacle problems
  • A priori estimates
  • Nearly linear growth

Mathematics Subject Classification

  • Primary 49N60
  • 35J85
  • Secondary 35B65
  • 35B45
  • 49J40