Global gradient estimates for the mixed local and nonlocal problems with measurable nonlinearities

A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón–Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón–Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical \((-\Delta _p)^1\) ${(-{\mathrm{\Delta }}_p)}^1$ -Laplace operator with \(1<p<\infty \) $1<p<\infty$ , is measurable in one variable and has a small BMO assumption for the other variables, while the singular kernel associated with the nonlocal \((-\Delta _q)^s\) ${(-{\mathrm{\Delta }}_q)}^s$ -Laplace operator with \(0<s<1<q<\infty \) $0<s<1<q<\infty$ is merely measurable, and that the boundary of the domain is sufficiently flat in Reifenberg sense, we prove that the gradient of the solution is as integrable as that of the associated non-homogeneous term in the divergence form. Our regularity theory reported here could be a nice addition for higher integrability of the gradient to the Hölder continuity of the gradient in De Filippis and Mingione (Math Ann https://doi.org/10.1007/s00208-022-02512-7, 2022) regarding mixed local and nonlocal problems.