Improved regularity for the \(p\)-Poisson equation

In this paper we produce new, optimal, regularity results for the solutions to \(p\)-Poisson equations. We argue through a delicate approximation method, under a smallness regime for the exponent \(p\), that imports information from a limiting profile driven by the Laplace operator. Our arguments contain a novelty of technical interest, namely a sequential stability result; it connects the solutions to \(p\)-Poisson equations with harmonic functions, yielding improved regularity for the former. Our findings relate a smallness regime with improved \(\mathcal{C}^{1,1-}\)-estimates in the presence of \(L^\infty\)-source terms.