A natural local boundedness result for solutions of elliptic–parabolic equations

We define a homogeneous De Giorgi class of order [Formula: see text] that encompasses solutions to evolution equations of the forms [Formula: see text] and [Formula: see text], where [Formula: see text] almost everywhere and [Formula: see text] is an appropriate elliptic operator. For functions belonging to this class, we prove an inhomogeneous boundedness result. As a consequence, and this is the main result of the paper, we establish local boundedness for solutions of strongly degenerate parabolic (or elliptic–parabolic) equations.