Riesz potential estimates for non-linear elliptic obstacle problems

Technical Summary: Riesz Potential Estimates for Non-linear Elliptic Obstacle Problems

Problem Statement
This paper investigates the regularity of solutions to a class of non-homogeneous nonlinear elliptic obstacle problems involving measure data. The primary equation under consideration is:
$$-\text{div}(a(x, u, Du)) = \mu \quad \text{in } \Omega,$$
subject to the obstacle constraint $u \geq \psi$ almost everywhere in a bounded open set $\Omega \subseteq \mathbb{R}^n$ ($n \geq 2$). Here, $\mu$ is a signed Radon measure with finite total variation ($\mu \in \mathcal{M}_b(\Omega)$), and the nonlinearity $a(x, z, \eta)$ depends on the spatial variable $x$, the solution $z$, and the gradient $\eta$.

The operator $a$ satisfies standard ellipticity and growth conditions for a $p$-growth exponent $p \geq 2$. Crucially, the authors impose two specific continuity assumptions on $a$:

1. Dini-continuity in $x$: The dependence on the spatial variable satisfies a Dini condition, ensuring a modulus of continuity $\omega$ such that $\int_0^R \frac{\omega(\rho)}{\rho} d\rho < \infty$.
2. Hölder continuity in $z$: The dependence on the solution variable $z$ satisfies a natural Hölder condition with exponent $\sigma$.

The obstacle function $\psi$ is assumed to belong to $W^{2,1}(\Omega) \cap W^{1,p}(\Omega)$ with specific integrability conditions on its derivatives. The solution $u$ is defined as a limit of approximating solutions to regularized problems, a concept established in prior literature for measure data.

Methodology
The authors employ a perturbative approach based on comparison estimates and excess decay arguments. The core methodology involves the following steps:

1. Approximation and Comparison: The solution $u$ is compared against a sequence of auxiliary functions ($w_1, \dots, w_5$) defined on nested balls. These functions solve related homogeneous or simplified obstacle problems where the measure data and obstacle effects are progressively removed or frozen.

   • $w_1$ solves the obstacle problem with zero right-hand side.
   • $w_2$ and $w_3$ handle the obstacle term and the homogeneous equation, respectively.
   • $w_4$ and $w_5$ involve freezing the coefficients in the solution variable and spatial variable, respectively.

2. Comparison Estimates: The authors derive $L^q$ estimates for the differences between the gradients of these auxiliary functions. These estimates quantify the error introduced by the measure data $\mu$, the obstacle $\psi$, and the nonlinearity's dependence on $u$. The estimates rely on the properties of the $p$-Laplacian type structure and the specific continuity assumptions on $a$.

3. Excess Decay and Iteration: By combining the comparison estimates with a decay lemma for the excess (oscillation of the gradient), the authors establish an iterative decay estimate. This process utilizes the Dini condition on the modulus of continuity $\omega$ and the smallness of the measure data and obstacle terms in specific integral norms.

4. Normalization and Scaling: To handle the global dependence on the solution $u$ in the nonlinearity, the authors utilize a normalization technique. They rescale the problem to normalize the gradient magnitude, allowing the application of local regularity results uniformly.

5. Potential Estimates: The iterative decay estimates are integrated to derive pointwise bounds in terms of Riesz potentials. The authors distinguish between the contribution of the measure data $\mu$ and the obstacle term $\psi$ (denoted via a term $D\Psi$).

Key Contributions and Results
The paper extends the theory of potential estimates from nonlinear elliptic equations to the more complex setting of obstacle problems where the nonlinearity explicitly depends on the solution itself. The main results are:

• Theorem 1.2 (Pointwise Gradient Estimate): The authors establish a pointwise gradient estimate for the solution $u$ at a Lebesgue point $x_0$. The estimate bounds $|Du(x_0)|$ by the sum of:

  • An integral average of $|Du|$ over a ball.
  • The Riesz potential of the measure data, $I_{|\mu|}^1(x_0, R)$.
  • A term involving the Riesz potential of the obstacle data, $I_{[\psi]}^1(x_0, R)$.
  • A term accounting for the nonlinearity's dependence on the solution, involving the constant $\Gamma$, the radius $R$, and the exponent $\sigma$.
    The estimate takes the form:
    $$|Du(x_0)| \leq c \left( \text{Average} + (I_{|\mu|}^1 + I_{[\psi]}^1)^{\frac{1}{p-1}} + \text{Nonlinear Correction Term} \right).$$

• Theorem 1.3 (VMO Regularity): Under the assumption that the Riesz potentials of the measure and obstacle data are bounded and vanish in the limit as the radius goes to zero, the gradient $Du$ is shown to be in the space of functions with Vanishing Mean Oscillation (VMO) locally.

• Theorem 1.4 (Gradient Continuity): If the Riesz potentials of the measure and obstacle data vanish at the limit (a stronger condition than in Theorem 1.3), the gradient $Du$ is proven to be continuous in the domain.

Significance and Claims
The paper positions itself as an extension of existing potential theory for obstacle problems. While previous works (e.g., Scheven [24, 25]) established gradient estimates in terms of Wolff potentials for obstacle problems, and others (Byun, Song, Youn [2, 3]) introduced Riesz potential estimates for constant-coefficient obstacle problems, this work addresses the gap where the nonlinearity depends on the solution variable $u$.

The authors claim that their results provide a comprehensive tool for obtaining regularity estimates for these general problems. By deriving estimates specifically in terms of Riesz potentials (which are simpler and more explicit than Wolff potentials in the case $p=2$ and often more tractable for $p \neq 2$), the paper offers a refined understanding of how the regularity of the solution depends on the integrability of the measure data and the smoothness of the obstacle. The results generalize previous findings to a setting with variable coefficients and solution-dependent nonlinearities, confirming that the regularity of the gradient is controlled by the Riesz potentials of the data and the obstacle, provided the nonlinearity satisfies the specified Dini and Hölder conditions.