In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type
| $$ |
Here $$ is the set of admissible functions z ∈ u0 + W$$(Ω) for a given u0 ∈ W$$(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝ$$, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.
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Keywords: Variational inequalities, obstacle problems, local boundedness, local Lipschitz continuity
@article{COCV_2021__27_1_A21_0,
author = {Caselli, M. and Eleuteri, M. and Passarelli di Napoli, A.},
title = {Regularity results for a class of obstacle problems with \protect\emph{p}, \protect\emph{q}\ensuremath{-}growth conditions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021017},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021017/}
}
TY - JOUR AU - Caselli, M. AU - Eleuteri, M. AU - Passarelli di Napoli, A. TI - Regularity results for a class of obstacle problems with p, q−growth conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021017/ DO - 10.1051/cocv/2021017 LA - en ID - COCV_2021__27_1_A21_0 ER -
%0 Journal Article %A Caselli, M. %A Eleuteri, M. %A Passarelli di Napoli, A. %T Regularity results for a class of obstacle problems with p, q−growth conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021017/ %R 10.1051/cocv/2021017 %G en %F COCV_2021__27_1_A21_0
Caselli, M.; Eleuteri, M.; Passarelli di Napoli, A. Regularity results for a class of obstacle problems with p, q−growth conditions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 19. doi: 10.1051/cocv/2021017
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