Pointwise estimates of potentials for higher-order capacities

Abstract

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)⩜W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)−kf(x)∈W _p ^m (D)⩜W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F