2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217466It is shown that, for the bi-harmonic equation, an optimal regularity criterion of the vertex of typical paraboloids can be expressed in terms of Osgood-Dini integral conditions of Petrovskii's type for the heat equation derived in 1934. Some extensions of the first Fourier coefficent method to other PDEs are discussed. A survey on boundary point regularity for elliptic and parabolic PDEs is enclosed.Analysis of PDEs35K55, 35K65text