Optimal constant for the trace inequality in $BV$ for domains with corners

1. Problem Statement and Motivation

The paper addresses a fundamental problem in the Calculus of Variations concerning functions of bounded variation (BV) on domains $\Omega \subset \mathbb{R}^n$ with non-smooth boundaries.

• Context: The trace operator $\text{Tr}: BV(\Omega) \to L^1(\partial\Omega)$ allows the definition of boundary values for BV functions. A standard trace inequality states that there exists a constant $C$ such that:
  $$\int_{\partial\Omega} |\text{Tr}(u)| \, d\mathcal{H}^{n-1} \leq C \left( \int_{\Omega} |u| \, dx + |Du|(\Omega) \right)$$
  where $|Du|(\Omega)$ is the total variation of the distributional derivative.
• The Specific Challenge: In variational problems involving both bulk and boundary energies (e.g., minimizing $F(u) = |Du|(\Omega) + \alpha \int_{\partial\Omega} |\text{Tr}(u)| \, d\mathcal{H}^{n-1}$), the existence of a minimizer depends critically on the sign and magnitude of $\alpha$. Specifically, if $\alpha < 0$, a solution exists only if the boundary term is controlled by the total variation term.
• The Optimal Constant: The authors focus on determining the optimal constant $Q_{\partial\Omega}$ (denoted as $q_{\partial\Omega}(x)$ locally) such that:
  $$\int_{\partial\Omega} |\text{Tr}(u)| \, d\mathcal{H}^{n-1} \leq (Q_{\partial\Omega} + \epsilon)|Du|(\Omega) + C_\epsilon \int_{\Omega} |u| \, dx$$
  For smooth ($C^1$) boundaries, it is known that $Q_{\partial\Omega} = 1$. However, for domains with corners or singularities, this constant can be strictly greater than 1. The paper aims to explicitly calculate this constant for a specific class of singularities (cones) in dimensions $n \geq 2$.

2. Methodology

The authors employ tools from Geometric Measure Theory to transform the analytic problem into a geometric variational problem.

• Reduction to Cones: By analyzing the local behavior of the boundary near a singularity $x \in \partial\Omega$, the problem is reduced to studying a cone $C \subset \mathbb{R}^n$ that approximates the domain near $x$. The local constant $q_{\partial\Omega}(x)$ is equivalent to the supremum of a ratio over sets of finite perimeter inside the cone:
  $$\alpha(C) = \sup \left\{ \frac{1}{P(E; C)} \int_{\partial C} \text{Tr}(\mathbf{1}_E) \, d\mathcal{H}^{n-1} : E \subset C, |E|>0 \right\}$$
• Gauss-Green Measure and Divergence Theorem: The proof relies heavily on the generalized Gauss-Green theorem for sets of finite perimeter. By choosing a specific constant vector field $T = (0, \dots, 0, 1)$, they relate the boundary integral to the interior perimeter via:
  $$\int_{\partial^* E \cap C} T \cdot \nu_E \, d\mathcal{H}^{n-1} = - \int_{\partial^* E \cap \partial C} T \cdot \nu_C \, d\mathcal{H}^{n-1}$$
• Geometric Characterization of the Normal: The core of the methodology involves analyzing the geometry of the cone's boundary $\partial C$. The authors identify a condition where the inner product $T \cdot \nu_C$ is constant on the boundary. This occurs when the base of the cone admits an inscribed sphere (or a specific tangency condition with a sphere).
• Existence vs. Maximization: The paper investigates whether a set $E$ actually attains this supremum. They demonstrate that due to the scaling invariance of the cone, minimizing sequences may converge to a set of zero volume (a "vanishing" solution), meaning a maximizer might not exist in the strict sense for certain geometries, even if the optimal constant is well-defined.

3. Key Contributions and Results

A. Explicit Formula for the Optimal Constant

The main result (Theorem 2 and Theorem 19) provides an explicit formula for the optimal constant $\alpha(C)$ for cones whose base $G(f) \subset \mathbb{R}^{n-1}$ satisfies a specific geometric condition: the existence of a sphere $S_R(\bar{x})$ such that every hyperplane tangent to the base $G(f)$ is also tangent to the sphere.

Under this condition, the optimal constant is:
$$\alpha(C) = \frac{\sqrt{1 + R^2}}{R} = \sqrt{1 + \frac{1}{R^2}}$$
where $R$ is the radius of the inscribed sphere in the base of the cone.

B. Characterization of Maximizers

The paper characterizes the sets $E$ that achieve this supremum (if they exist). A set $E$ is a solution if and only if it is the intersection of the cone $C$ with a half-space defined by a plane orthogonal to the axis of the cone:
$$E = C \cap \{ x \in \mathbb{R}^n : x \cdot \bar{x} \leq t \}$$
for some $t > 0$. This confirms the heuristic that the "optimal" shape cutting the cone is a flat slice perpendicular to the cone's axis.

C. Generalization to 2D and 3D

• Dimension $n=2$: The result generalizes the known 2D case where the optimal constant depends on the angle of the corner. For a 2D cone (an angle), the "inscribed sphere" is simply the distance between the two rays, and the formula recovers the known result where the optimal cut is orthogonal to the angle bisector.
• Dimension $n \geq 3$: The paper extends this to higher dimensions. It shows that for a cone with a square base (which has an inscribed circle), the constant is determined by the radius of that circle. However, it notes that for shapes like a non-square rectangle (which lacks an inscribed circle tangent to all sides), the theorem does not directly apply, highlighting the limitations of the current geometric condition.

D. Non-Existence of Maximizers

The authors provide a counter-example (the "book cone" with a 90-degree fold) where the supremum is $\sqrt{2}$, but no set of positive volume achieves this value. The maximizing sequence consists of prisms with height $h \to 0$, which converges to a set of measure zero. This clarifies that while the constant is well-defined, the existence of an optimal set is not guaranteed for all singularities.

4. Significance

• Resolution of a Specific Open Problem: The paper provides the first explicit calculation of the trace inequality constant for a broad class of singular domains in dimensions $n \geq 3$, moving beyond the known $C^1$ case ($Q=1$).
• Geometric Insight: It establishes a direct link between the geometry of the boundary singularity (specifically the radius of an inscribed sphere in the cross-section) and the analytic constant in the trace inequality.
• Implications for Variational Problems: The results are crucial for determining the existence of minimizers in free boundary problems and image processing models (like the Mumford-Shah functional) where boundary energies are penalized. Knowing the precise threshold for $\alpha$ allows researchers to determine when a problem is well-posed.
• Methodological Rigor: The paper rigorously addresses the subtle issue of existence versus asymptotic behavior in scaling-invariant problems, distinguishing between the value of the constant and the existence of an optimizer.

In summary, Cristoferi and Van Der Gulik successfully determine that for cones with bases admitting an inscribed sphere, the optimal trace constant is $\sqrt{1+R^{-2}}$, achieved by planar cuts orthogonal to the cone's axis, thereby bridging geometric measure theory and the analysis of singular variational problems.