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We investigate an optimalW   1  ,   p  ⁢   (  ⋅  )   ⁢  q${W^{1,p(\,\cdot\,)q}}$   -regularity theory for a nonlinear elliptic obstacle problem with nonstandard growthp  ⁢   (  ⋅  )${p(\,\cdot\,)}$   . With a sufficient small log-Hölder constant onp  ⁢   (  ⋅  )${p(\,\cdot\,)}$   , under a suitable smallness condition in BMO on the nonlinearity and under a sufficient flatness condition on the boundary of the domain, we establish a global Calderón–Zygmund estimate for such an irregular obstacle problem by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the inhomogeneous term in the variable exponent spaceL   p  ⁢   (  ⋅  )   ⁢  q${L^{p(\,\cdot\,)q}}$    for everyq  ∈   (  1  ,  ∞  )${q\in(1,\infty)}$   .

Global gradient estimates for nonlinear obstacle problems with nonstandard growth