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

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

Problem Statement
The paper addresses the development of a nonlinear potential theory for elliptic equations of the form $-\text{div} A(x, \nabla u) = \mu$ in a bounded domain $\Omega \subset \mathbb{R}^n$, where the operator $A$ exhibits general Orlicz growth. The central challenge is the absence of homogeneity and scaling invariance. Unlike the classical $p$-Laplace setting where $A(x, \lambda \xi) = \lambda |\lambda|^{p-2} A(x, \xi)$, the operators considered here satisfy growth conditions governed by a doubling function $G$ (specifically $A(x, \xi) \cdot \xi \approx G(|\xi|)$) without assuming any scaling property. This lack of scaling prevents the direct application of classical tools from nonlinear potential theory, such as rescaling arguments for solutions or capacitary potentials, and creates a gap in the equivalence theory between variational solutions (renormalized solutions) and potential-theoretic objects (superharmonic functions) for measure data problems.

Methodology
To overcome the lack of scaling, the authors construct a new, scaling-free framework. The methodology proceeds through several interconnected stages:

1. Regularity of Obstacle Problems: The foundation is laid by establishing global Hölder regularity for solutions to obstacle problems under the general Orlicz growth conditions. This result is critical as it provides the necessary regularity to construct and approximate balayages without relying on scaling.
2. Scale-Free Balayage Theory: The authors develop a theory of balayage (sweeping) and réduite that does not depend on homogeneity. They prove that the balayage of a function relative to a set coincides with the solution to a corresponding obstacle problem.
3. Riesz Measures and Integrability: Using the balayage theory, the paper establishes the integrability properties of $A$-superharmonic functions, specifically showing that $g(|\nabla u|) \in L^1_{\text{loc}}$. This allows for the definition of a unique nonnegative Radon measure, the Riesz measure $\mu[u]$, associated with any $A$-superharmonic function $u$, satisfying the equation in the sense of distributions.
4. Capacitary Estimates: The authors derive quantitative estimates for the Riesz measures of balayages corresponding to different constant obstacles. These estimates are used to prove that $A$-polar sets (sets where superharmonic functions can blow up) have zero $G$-capacity.
5. Quasicontinuity: Leveraging the capacity estimates, the paper proves that $A$-superharmonic functions are $G$-quasicontinuous. This property ensures that integrals against diffuse measures are well-defined independently of the chosen representative.
6. Equivalence Proof: Finally, the authors combine the potential-theoretic framework with localization arguments near the singular part of the measure. They utilize Wolff potential estimates and the concentration property of the singular measure on the set $\{u = \infty\}$ to bridge the gap between the distributional formulation and the renormalized formulation.

Key Contributions and Results
The paper presents the following main results:

• Theorem 1.3 (Capacitary Negligibility): Every $A$-polar set has zero $G$-capacity. This extends a fundamental principle of nonlinear potential theory to the non-homogeneous Orlicz setting.
• Theorem 1.4 (Quasicontinuity): If $u$ is an $A$-superharmonic function, then $u$ is $G$-quasicontinuous in $\Omega$.
• Theorem 1.6 (Existence of Representative): Every renormalized solution to the measure data problem admits an $A$-superharmonic representative that coincides with the solution almost everywhere.
• Theorem 1.7 (Renormalized Structure): Every $A$-superharmonic function $u$ is a renormalized solution to the equation with datum equal to its Riesz measure $\mu[u]$.
• Main Equivalence: Consequently, the classes of $A$-superharmonic functions and renormalized solutions coincide (up to the choice of the lower semicontinuous representative).

Significance and Claims
The paper claims to provide the first scaling-free potential theory for elliptic equations with general Orlicz growth. Its primary significance lies in extending the classical equivalence theory between superharmonic functions and renormalized solutions—previously established only in the homogeneous $p$-growth setting—to the general Orlicz framework.

The authors emphasize that this result is new even for power-growth operators ($G(t) = t^p/p$) when the homogeneity assumption (1.2) is dropped. In the classical setting, the equivalence relied heavily on scaling arguments and the capacitary negligibility of polar sets derived from homogeneous potential theory. By developing tools that do not require scaling invariance, the authors establish a "genuinely scale-free counterpart" to the classical equivalence theory.

The developed framework, including the balayage theory, Riesz measure analysis, and capacitary estimates, is presented not only as a means to prove the equivalence but also as a self-contained nonlinear potential theory of independent interest. It is intended to serve as a basis for further study of fine properties of solutions and measure data problems in non-homogeneous settings. The paper does not propose new applications or experimental proposals but focuses strictly on the theoretical development and the rigorous establishment of these equivalences.