Higher differentiability of solutions for a class of obstacle problems\n with variable exponents

2021/07/09 by Niccolò Foralli, Foralli, Niccolò, Giovanni Giliberti +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2107.04336

Abstract

In this paper we prove a higher differentiability result for the solutions to\na class of obstacle problems in the form\n \
labelobst-def0
min
left

int_
Omega F(x,Dw) dx : w
in\n
mathcalK
_psi(
Omega)
right
where \ψ\∈\nW^1,p(x)(\Ω) is a fixed function called obstacle and\n\K_\ψ= w \∈ W^1,p(x)₀(\Ω)+u₀: w \≥ \ψ , ,\n textnormala.e. in \Ω is the class of the admissible functions, for\na suitable boundary value u₀ . We deal with a convex integrand F which\nsatisfies the p(x)-growth conditions\n\
labelgrowth|
xi|^p(x)
le F(x,
xi)
le\nC(1+|
xi|^p(x)),
quad p(x)gt;1 n

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