Viscosity Solutions for First‐Order PDEs With Measurable Coefficients and Superlinear Gradient Growth

Inspired by a fundamental existence result in the theory of PDEs, we present a general existence theorem for viscosity solutions in the standard sense that is applicable to a wide class of partial differential equations. These equations are characterized by coefficients that are merely measurable, with no continuity assumptions imposed. To illustrate the scope and flexibility of the main result, we derive several propositions for specific families of PDEs, including examples that fall outside the reach of prior existence theorems with stronger regularity hypotheses.