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This paper deals with a parabolic-elliptic chemotaxis-consumption system withtensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$and Robin-type boundary conditions for $c$. The global existence of boundedclassical solutions is established in dimension two under general assumptionson tensor-valued sensitivity $S$. One of main steps is to show that $\nablac(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap \Omega)$ for every$x\in\overline{\Omega}$ and $t$ when $r$ is sufficiently small, which seems tobe of independent interest. On the other hand, in the case of scalar-valuedsensitivity $S=\chi(x,n,c)\mathbb{I}$, there exists a bounded classicalsolution globally in time for two and higher dimensions provided the domain isa ball with radius $R$ and all given data are radial. The result of the radialcase covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.

Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions