Higher order Hölder approximation by solutions of second order elliptic equations

Journal article, 2025

For a given second order elliptic operation L in a domain Ω⊂RN, N≥3, and a compact set K⊂Ω, order N-2-Ahlfors-David regular, we define the space HLr+ω(K) of continuous functions f(x),x∈K, admitting, for any δ>0, a local approximation in the δ-neighborhood of any point x∈K, with δrω(δ)-error estimate, by solutions of the equation Lu=0. For such functions, we prove the existence of a global approximation vδ on K with the same order of error estimate, by a solution of the same equation in a δ-neighborhood of K. A number of properties of these functions vδ and their derivatives are established.