Maximum principle for higher order elliptic operators with inertia in general domains and any dimension

It is well known how the Maximum Principle (MP) in general fails to hold for uniformly elliptic operators of order higher than two, even in smooth convex domains. In D. Cassani and A. Tarsia (2022) it was shown in dimension $N=2,3$, by establishing a new Harnack type inequality, that the validity of the positivity preserving property can be restored when lower order derivatives are taken into account as a perturbation of the higher order differential operator. The restriction to the dimension was due to regularity issues which we develop here, extending the validity of the MP to any dimension and fairly general domains. Moreover, we show that the presence of inertial terms affects the range of the perturbation parameter, providing a balance between the positivity restoring effect of lower order derivatives and the mass energy. The method provided here is flexible with respect to the form of differential operators involved and thus suitable to be further extended to other classes of operators than just elliptic.