Newtonian capacity and quasi-balayage

Abstract

For various applications (including: regularity study of free boundaries, and approximation in the mean of harmonic functions on unbounded domains by rapidly decreasing ones) it is desirable to have an estimate, valid for an arbitrary smooth function φ of compact support in IRn, bounding sup |φ| in the unit ball B in terms of sup |∆φ| on IR^n. Such an estimate cannot exist with no further assumptions on φ, but in the predecessor of this paper it was shown that if φ vanishes on a subset E of B with volume |E| > 0, such a bound holds , and uniformly with respect to all sets E with |E| not less than any prescribed positive constant. In the present paper an analogous estimate is obtained for φ which |φ(0)| ≤ 1, and grad(φ) vanishes on a subset E of B whose Newtonian capacity exceeds a positive constant. Various applications are given. The derivation of the basic estimate involves ideas which may have some independent interest, in particular “quasi-balayage”, the sweeping-out of a measure from its support to some other compact set such that the potentials of the two measures, while not equal on a neighborhood of infinity as required by (true) balayage, are asymptotically equal to some prescribed degree.

Keywords

• balayage
• newtonian capacity
• a priori estimates
• approximation of harmonic functions by potentials