Partial regularity of the gradient for subsolutions

1. Problem Statement and Motivation

The paper addresses the regularity of the gradient ($\nabla u$) for bounded subharmonic functions (and more general subsolutions) in the context of free boundary problems and Dirichlet problems.

• Core Question: Under what minimal geometric conditions on the level sets of a bounded subharmonic function $u$ (where $\Delta u \ge 0$) is the gradient $\nabla u$ upper semi-continuous?
• Context: While the differentiability of solutions up to the boundary is a well-studied topic, the specific question of the upper semi-continuity (USC) of the gradient at the boundary or on level sets has been less explored. Previous work by the authors in convex settings suggested this property holds, but a general framework was missing.
• Key Obstacles:
  • Semi-continuity Direction: The authors argue for upper semi-continuity rather than lower. In non-convex domains (e.g., "zig-zag" boundaries), the gradient can blow up at inward corners when approached non-tangentially, violating lower semi-continuity, but upper semi-continuity remains a viable target.
  • Definition of Gradient: At points where $u$ is not $C^1$, the classical gradient is undefined. The paper proposes a definition based on integral means that coincides with the classical gradient where smooth.
  • Counter-examples: The paper cites examples (e.g., Alt-Caffarelli) where gradients behave differently depending on the direction of approach to a singularity, necessitating strong one-sided geometric assumptions to obtain regularity.

2. Methodology

The authors employ a combination of geometric analysis and monotonicity formulas to establish their results.

A. Geometric Assumption: Exterior Touching

The central hypothesis is a geometric condition on the super-level sets $D^+_y = \{x : u(x) > u(y)\}$.

• Condition: For every point $y$, the set $D^+_y$ can be touched from the exterior at $y$ by a connected domain $K_y$ whose boundary $\partial K_y$ is uniformly $C^{1, \text{Dini}}$ in a neighborhood of $y$.
• Implication: This "exterior touching" condition allows the construction of barrier functions, proving that $u$ is uniformly Lipschitz continuous.

B. The ACF Monotonicity Formula

The primary analytical tool is the Alt-Caffarelli-Friedman (ACF) monotonicity formula.

• For two non-negative functions $h_+, h_-$ with disjoint supports and $\Delta h_\pm \ge 0$, the functional:
  $$I(r, y, h_+, h_-) = I(r, y, h_+) I(r, y, h_-)$$
  is non-decreasing in $r$, where $I(r, y, h) = \frac{1}{r^2} \int_{B_r(y)} \frac{|\nabla h|^2}{|x-y|^{n-2}} dx$.
• The authors utilize this formula to define the gradient magnitude at points where $u$ is not $C^1$.

C. Definition of the Gradient

Since $u$ is Lipschitz, the authors define $|\nabla u(y)|$ via the limit of the monotonicity functional:
$$|\nabla u(y)|^2 := \frac{1}{c_0} \lim_{r \to 0} I(r, y, u_y)$$
where $u_y = \max(u - u(y), 0)$ and $c_0$ is a dimensional constant. This definition is shown to be independent of the specific auxiliary harmonic function used for comparison and coincides with the classical gradient at $C^1$ points.

D. Blow-up and Asymptotic Analysis

To prove upper semi-continuity, the authors use a blow-up argument:

1. Assume $\nabla u(y_0) \neq 0$.
2. Rescale the function around $y_0$.
3. Show that the limit function $u_0$ satisfies a constant-coefficient PDE and, by the strong form of the monotonicity formula, must be linear (or zero) in a half-space.
4. This asymptotic behavior ($u(x) \approx c_1(x_1 - y_{0,1})_+$) allows the comparison of gradients at nearby points to the gradient at the limit point.

3. Key Contributions and Results

Main Theorem (Theorem 2.1)

Let $u$ be a bounded function satisfying $\Delta u \ge -1$ in $B_1$. If the super-level sets of $u$ satisfy the uniform exterior $C^{1, \text{Dini}}$ touching condition for all points in a subdomain, then $\nabla u$ is upper semi-continuous in the interior.

Generalizations

1. Poisson Equation: The result extends to $\Delta u \ge -1$ using a modified monotonicity formula (Caffarelli-Jerison-Kenig).
2. Parabolic/Time-Dependent Operators: The result holds for the heat operator and general divergence-type operators $F[u] = \text{div}(A\nabla u) + b\cdot\nabla u + cu - d\partial_t u$, provided $\partial_t u$ is bounded from below and coefficients satisfy Dini continuity.
3. Dirichlet Problem (Theorem 3.1): The authors extend the result to solutions of the Dirichlet problem $Lu = f$ in a domain $\Omega$. If the domain boundary $\partial \Omega$ satisfies the exterior $C^{1, \text{Dini}}$ touching condition, then the spatial gradient $\nabla u(x, t)$ is upper semi-continuous for fixed $t$.

Technical Lemmas

• Lemma A.1: Proves that the exterior touching condition implies the solution is uniformly Lipschitz via a barrier argument.
• Lemma A.3: Establishes the asymptotic expansion of $u$ near a point where the gradient is non-zero, showing it behaves like a linear function in a half-space.

4. Significance and Impact

• New Regularity Criterion: The paper identifies a specific geometric condition (exterior $C^{1, \text{Dini}}$ touching) that is sufficient to guarantee the upper semi-continuity of the gradient, a property previously known only in highly symmetric (convex) settings.
• Robust Definition of Gradient: It provides a rigorous, intrinsic definition of the gradient for non-smooth subharmonic functions that is stable under limits and consistent with classical derivatives.
• Broad Applicability: By utilizing the ACF monotonicity formula, the results are not limited to the Laplacian but extend to a wide class of linear and nonlinear operators, including time-dependent (parabolic) equations.
• Free Boundary Theory: The findings are crucial for free boundary problems (like the Bernoulli problem), where understanding the behavior of the gradient at the interface is essential for proving the regularity of the free boundary itself.

In summary, the paper bridges the gap between geometric conditions on level sets and the analytic regularity of gradients, proving that "nice" exterior geometry prevents the gradient from oscillating wildly, thereby ensuring upper semi-continuity.