Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth

Technical Summary: Wolff Potential Estimates for Elliptic Obstacle Problems with Generalized Orlicz Growth

Problem Statement
This paper investigates elliptic obstacle problems involving measure data within the framework of generalized Orlicz spaces. The primary equation under consideration is:
$$-\text{div}\left( \frac{g(x, |Du|)}{|Du|} Du \right) = \mu \quad \text{in } \Omega,$$
subject to the obstacle constraint $u \geq \psi$ almost everywhere in a bounded domain $\Omega \subseteq \mathbb{R}^n$ ($n \geq 2$). Here, $\mu$ is a signed Radon measure with finite total variation, and $g(x, t)$ represents a nonlinearity satisfying generalized Orlicz growth conditions. This setting encompasses specific cases such as $p(x)$-growth, perturbed variable exponent growth, and double-phase growth. The obstacle function $\psi$ is assumed to belong to the Musielak-Orlicz-Sobolev space $W^{1,G}(\Omega)$, without requiring higher regularity assumptions (e.g., $W^{2,1}$) on the obstacle that were present in prior literature.

Methodology
The authors employ a unified approach to handle the elliptic obstacle problems, avoiding the need for separate treatments for the cases $p < 2$ and $p \geq 2$. The methodology proceeds through the following stages:

1. Existence Theory: The existence of solutions in the Musielak-Orlicz space is established using approximation techniques. The authors consider a sequence of approximating problems with regularized data $f_i \in L^1(\Omega) \cap (W^{1,G}(\Omega))'$ converging to the measure $\mu$. By utilizing the reflexivity of the space and compactness arguments (Banach-Alaoglu and Dunford-Pettis theorems), they prove the existence of a limit of approximating solutions.
2. Comparison Estimates: A multi-step comparison argument is utilized to bridge the gap between the solution of the inhomogeneous obstacle problem and solutions to homogeneous equations. The process involves:
   • Comparing the solution $u$ with a solution $w_1$ to a homogeneous obstacle problem with the same boundary data.
   • Comparing $w_1$ with a solution $w_2$ to a problem with regularized coefficients (using a regularized Orlicz function $\tilde{G}$).
   • Comparing $w_2$ with a solution $w_3$ to a related elliptic equation without the obstacle constraint.
   • Comparing $w_3$ with a solution $w_4$ to a homogeneous equation with frozen coefficients.
3. Excess Decay and Potential Estimates: Leveraging the comparison estimates, the authors derive excess decay estimates for the gradients. These estimates are then iterated to obtain pointwise bounds. The core of the analysis relies on the Wolff potential and restricted fractional maximal functions to quantify the influence of the measure data $\mu$ and the obstacle $\psi$ on the regularity of the solution.

Key Contributions and Results

• Existence of Solutions (Theorem 1.13): The paper establishes the existence of at least one approximable solution to the obstacle problem $OP(\psi; \mu)$ in the Musielak-Orlicz space $T^{1,G}_h(\Omega)$, under minimal structural assumptions on the growth function $G$ (specifically conditions (aInc)$_p$, (aDec)$_q$, (A0), and (A1)).
• Pointwise Gradient Estimates (Theorem 1.15): The authors derive pointwise estimates for the gradients of solutions in terms of the Wolff potential $W^\mu_{\beta, p}$ and the restricted sharp fractional maximal function $M^\#_{\alpha, R}$. These estimates explicitly account for the measure data $\mu$ and the regularity of the obstacle $\psi$. The estimates hold under the assumption that the obstacle satisfies a Dini-BMO regularity condition.
• Oscillation and $C^{1,\alpha}$-Regularity (Theorem 1.16): By refining the pointwise estimates, the paper derives oscillation estimates for the gradients. Specifically, it proves that if the measure data and the obstacle satisfy appropriate integrability and regularity conditions (including Dini-Hölder regularity for the modulus of continuity), the solution $u$ possesses $C^{1,\alpha}$ regularity. The gradient estimates are expressed as:
  $$|Du(x) - Du(y)| \leq C |x-y|^\alpha \left( \text{Integral terms involving } Du, \mu, \text{ and } \psi \right).$$
  The bounds involve the Wolff potential of the measure data and integral terms related to the obstacle's gradient.

Significance and Claims
The paper claims to extend previous findings in the field, particularly those in [50, 51], by removing the requirement for higher regularity conditions on the obstacle function $\psi$. Previous works often assumed $\psi \in W^{1,G}(\Omega) \cap W^{2,1}(\Omega)$ to facilitate the unified treatment of the measure $\mu$ and the obstacle term $D\Psi$. This work demonstrates that such higher regularity is not necessary for establishing Wolff potential estimates and $C^{1,\alpha}$ regularity.

Furthermore, the authors emphasize the generality of their framework, which covers Orlicz, variable exponent, and double-phase growth conditions under a single set of assumptions. A significant technical advantage highlighted is the unified treatment of the cases $p < 2$ and $p \geq 2$, which eliminates the need for distinct analytical approaches for different growth regimes. The results provide a comprehensive potential-theoretic description of the regularity of solutions to elliptic obstacle problems with measure data in the broad setting of generalized Orlicz growth.