Existence of free boundaries for overdetermined value problems: Sharp conditions, regularity, and physical applications

1. Problem Statement

The paper investigates the existence of free boundaries for Overdetermined Value Problems (ODVPs) involving the Laplacian and the bi-Laplacian operators. The study focuses on finding an open domain $\Omega$ (strictly containing a fixed convex core $C$) such that the solution to a specific PDE satisfies both a Dirichlet condition and a non-constant Neumann-type condition on the free boundary $\partial \Omega$.

Two primary problems are analyzed:

1. The Quadrature Surfaces Problem (QS):
   Find $\Omega \supset C$ such that:
   $$\begin{cases} -\Delta u_\Omega = f & \text{in } \Omega, \\ u_\Omega = 0 & \text{on } \partial \Omega, \\ |\nabla u_\Omega| = g & \text{on } \partial \Omega. \end{cases}$$
   Context: Models electrostatics (prescribed electric field magnitude) and fluid dynamics.

2. The Bi-Laplacian Problem (B):
   Find $\Omega \supset C$ such that:
   $$\begin{cases} \Delta^2 v_\Omega = f & \text{in } \Omega, \\ v_\Omega = \Delta v_\Omega = 0 & \text{on } \partial \Omega, \\ |\nabla v_\Omega| \cdot |\nabla(\Delta v_\Omega)| = g & \text{on } \partial \Omega. \end{cases}$$
   Context: Models the optimal design of simply supported thin plates under Kirchhoff-Love theory, where the condition represents a pointwise control of mechanical work density (product of slope and shear force).

Key Novelty: Unlike previous works that often assumed constant boundary data ($g \equiv k$), this paper addresses the case where $g$ is a non-constant function, which is more realistic for physical applications.

2. Methodology

The authors employ a unified approach combining variational methods, geometric analysis, and classical integral inequalities.

• Variational Framework:

  • For the Laplacian, they minimize the functional $J_{f,g}(\Omega) = \int_\Omega (|\nabla u_\Omega|^2 - 2fu_\Omega) dx + \int_\Omega g^2 dx$.
  • For the bi-Laplacian, they minimize $F_g(\Omega) = \int_\Omega (\frac{1}{2}u_\Omega^2 - g^2) dx$, where $u_\Omega = -\Delta v_\Omega$.
  • The existence of minimizers is established using the direct method of calculus of variations.

• Geometric Class (C-GNP):
  The analysis is conducted within the C-Geometric Normal Property (C-GNP) class of domains. This class ensures:

  • Compactness under Hausdorff convergence.
  • Continuity of state functions.
  • A specific geometric structure preventing "bubbles" and ensuring a unique "shadow" of the core $C$ on the free boundary.

• Regularity Analysis:
  The authors utilize the thickness function $d(c)$ (distance from $c \in \partial C$ to $\partial \Omega$) and the Wiener criterion to analyze the behavior of minimizers near contact points ($\partial \Omega \cap \partial C$). They prove that cusps are geometrically controlled (contained within a canonical cusp of revolution).

• Integral Inequalities:
  To derive existence conditions, the authors transform pointwise boundary conditions into integral inequalities using:

  • Cauchy-Schwarz, Hölder, and Hardy inequalities.
  • Pohozaev and Reilly identities.
  • Eigenvalue bounds and interpolation inequalities (Gagliardo-Nirenberg).

3. Key Contributions and Results

A. Sharp Existence Conditions

The paper establishes necessary and sufficient conditions for the existence of solutions, expressed as integral inequalities comparing the "source" inside $C$ to the "flux" on the boundary.

1. For the Laplacian (QS):

   • Necessary & Sufficient Condition:
     $$\int_C f \, dx > \int_{\partial C} g \, d\sigma$$
   • Physical Interpretation: The total source strength inside the core must strictly exceed the total flux that could be accommodated by the core's boundary. If the source is too strong for the core, the domain must expand to a larger $\Omega$ to distribute the flux.

2. For the Bi-Laplacian (B):

   • Sufficient Condition:
     $$\left( \int_{\partial C} \sqrt{g} \, d\sigma \right)^2 < \left( \int_C f \, dx \right) \left( \int_C u_C \, dx \right)$$
     where $u_C$ solves $-\Delta u_C = f$ in $C$ with $u_C=0$ on $\partial C$.
   • Physical Interpretation: The internal elastic strain energy of the plate (product of total load and integrated bending moment) must exceed the square of the averaged mechanical work prescribed on the boundary.

B. Regularity of Minimizers

• Theorem 2.7: Minimizers in the C-GNP class have a free boundary $\partial \Omega \setminus C$ that is $C^{1,\alpha}$ (smooth).
• Contact Set: Near the contact points where $\partial \Omega$ meets $\partial C$, the boundary is Wiener-regular. The free boundary meets the fixed boundary tangentially with at least quadratic detachment (the thickness function behaves like $d(c) \sim |c-c_0|^2$).
• This regularity justifies the use of shape derivative calculus even when the free boundary touches the fixed core.

C. Radial Case Analysis

The paper provides a rigorous analysis for the case where $C$ is a ball and data $f, g$ are radial.

• Symmetry: Solutions must be concentric balls.
• Explicit Conditions: The integral conditions reduce to algebraic inequalities involving the radius $R^*$.
• Comparison: The condition for the bi-Laplacian is shown to be significantly more restrictive than for the Laplacian.
  • Example: For a unit ball with constant data in 3D, the Laplacian allows $c < 1/3$, while the bi-Laplacian requires $c < 1/135$. This highlights the higher geometric rigidity of the biharmonic problem.

D. Stability and Symmetry Breaking

• Stability: Solutions are stable under small perturbations of $f$ and $g$ in $L^2$.
• Symmetry Breaking: If $C$ is not a ball and $g$ is non-constant, the solution $\Omega$ generally breaks symmetry (is not a dilation of $C$).

4. Physical Significance

The paper bridges abstract mathematical analysis with concrete physical interpretations:

• Electrostatics/Fluids: The Laplacian condition ensures that a conductor or fluid domain expands only if the internal charge/source cannot be balanced by the flux through the core's surface.
• Plate Theory: The bi-Laplacian condition represents a balance between the plate's ability to store elastic energy and the mechanical work required at the boundary. If the prescribed edge work is too high relative to the plate's stiffness, the plate must extend to "dilute" the stress.

5. Conclusion

This work extends the theory of overdetermined problems to non-constant boundary data, providing a unified variational framework for both second-order (Laplacian) and fourth-order (bi-Laplacian) operators. By establishing sharp integral conditions and proving $C^{1,\alpha}$ regularity for minimizers, the authors provide a robust theoretical foundation for shape optimization problems in potential theory, electromagnetism, and structural mechanics. The results demonstrate that while the Laplacian and bi-Laplacian problems share a similar variational structure, the bi-Laplacian imposes much stricter geometric constraints on the existence of solutions.