Orlicz Potential Theory: Balayage, Riesz Measures, and Very Weak Solutions

Abstract

We develop a nonlinear potential theory for elliptic equations with Orlicz growth under general monotonicity and growth conditions, without any homogeneity or scaling assumptions. The lack of scaling invariance prevents the use of many classical tools from nonlinear potential theory. To overcome this difficulty, we establish a new framework that includes global Hölder regularity for obstacle problems, a balayage theory, the construction and analysis of Riesz measures associated with superharmonic functions, the identification of capacitary potentials, capacitary estimates for polar sets, and the quasicontinuity of superharmonic functions. As an application of this theory, we prove that the classes of superharmonic functions and renormalized solutions to elliptic measure data problems coincide. This extends the classical equivalence theory from the homogeneous $p$-growth setting to general Orlicz growth and is new even for power-growth operators without homogeneity assumptions.