Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

Abstract

We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of [Formula: see text]-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in [Formula: see text] and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.