Surfing Regularity on Nonlinear Potentials

The representation formula for the Poisson equation gives an explicit expression of solutions in terms of the data, yielding zero- and first-order pointwise bounds via convolution with appropriate Riesz potentials. The mapping properties of these potentials provide sharp regularity transfer from data to solutions, giving a complete description of the regularity features of solutions. I will outline key aspects of nonlinear potential theory that reproduce this behavior for nonlinear elliptic PDEs, where representation formulae are unavailable, and trace their regularity theory back to that of the Poisson equation up to the C^{1} level. I will then present a novel potential-theoretic approach, altering a century old paradigm in nonlinear regularity theory, that resolves the longstanding problem of the validity of Schauder theory in nonuniformly elliptic PDEs.