Fully nonlinear elliptic PDEs in thin domains with oblique-Dirichlet mixed boundary conditions

1. Problem Statement

The paper investigates the asymptotic behavior of solutions to fully nonlinear elliptic partial differential equations (PDEs) defined on thin domains as the thickness of the domain approaches zero.

• Domain Geometry: The domain $\Omega_\varepsilon \subset \mathbb{R}^{N+1}$ is a "thin" layer collapsing to a bounded domain $\Omega \subset \mathbb{R}^N$. It is defined as:
  $$\Omega_\varepsilon = \{ (x, y) \in \Omega \times \mathbb{R} : \varepsilon g_-(x) < y < \varepsilon g_+(x) \}$$
  where $0 < \varepsilon \ll 1$, and $g_-, g_+$ are continuous functions with $g_- < g_+$.
• The Equation: The operator $F$ is of Bellman-Isaacs type (arising from stochastic differential games or control theory):
  $$F(D^2 u_\varepsilon, Du_\varepsilon, u_\varepsilon, z) = \inf_{\lambda \in L} \sup_{\mu \in M} \left( -\text{tr}(\sigma_{\lambda\mu}^\top \sigma_{\lambda\mu} D^2 u_\varepsilon) - b_{\lambda\mu} \cdot Du_\varepsilon + c_{\lambda\mu} u_\varepsilon - f_{\lambda\mu} \right) = 0$$
• Boundary Conditions: The problem features a mixed boundary condition:
  1. Oblique conditions on the top ($\partial_T \Omega_\varepsilon$) and bottom ($\partial_B \Omega_\varepsilon$) boundaries:
     $$\gamma_\pm \cdot Du_\varepsilon = \beta_\pm$$
  2. Dirichlet condition on the lateral boundary ($\partial_L \Omega_\varepsilon$):
     $$u_\varepsilon = \beta$$

The Core Challenge: Previous studies (e.g., [2]) required the operator $F$ to be strictly monotone in the solution variable $u$ (i.e., $c_{\lambda\mu} > 0$ strictly) to ensure existence and convergence. This paper addresses the more difficult case where strict monotonicity is relaxed to non-decreasing monotonicity ($c_{\lambda\mu} \geq 0$).

2. Methodology

The authors employ a combination of viscosity solution theory, barrier function construction, and coordinate transformation.

A. Relaxation of Monotonicity via Global Ellipticity

Since strict monotonicity ($c_{\lambda\mu} > 0$) is absent, the authors introduce a global ellipticity condition to compensate. They assume the existence of a function $s \in C^2(\Omega)$ such that the operator remains "elliptic enough" in a specific direction defined by $s$.

• Condition: There exists $s \in C^2(\Omega)$ such that for all $x \in \Omega$:
  $$\inf_{(\lambda, \mu)} \left| (Ds(x), -Ds(x)\gamma_0^\top(x)) A_{\lambda\mu}(x, 0) (Ds(x), -Ds(x)\gamma_0^\top(x))^\top \right| > 0$$
  where $A_{\lambda\mu} = \sigma_{\lambda\mu}^\top \sigma_{\lambda\mu}$.
• Significance: This condition ensures that even without strict monotonicity, the operator has sufficient coercivity to construct strict sub- and super-solutions. The paper provides a counter-example (Remark 1) showing that local ellipticity is insufficient; this global condition is necessary.

B. Construction of Strict Sub- and Super-Solutions (Lemma 2)

The proof of existence relies on Perron's method, which requires the construction of strict barriers (sub- and super-solutions) $\underline{u}_\varepsilon$ and $\overline{u}_\varepsilon$.

• Step 1 (Simplified Case): The authors first construct barriers assuming the oblique vector $\gamma_0 = 0$. They use an exponential ansatz involving the function $s$ (e.g., $e^{\alpha s(x)}$) to dominate the second-order terms.
• Step 2 & 3 (General Case - Coordinate Distortion): To handle the general case where $\gamma_0 \neq 0$, they introduce a change of variables (coordinate transformation):
  $$v_\varepsilon(z, y) = u_\varepsilon(z + y\gamma_0(z), y)$$
  This transformation "straightens" the oblique boundary conditions, effectively reducing the general problem to the simplified case ($\gamma_0=0$) analyzed in Step 1. The authors prove that this mapping is a $C^2$-diffeomorphism for small $\varepsilon$ and that the transformed equation retains the necessary ellipticity properties.

C. Convergence Analysis

Once the existence and uniform boundedness of solutions $u_\varepsilon$ are established, the authors analyze the limit as $\varepsilon \to 0$.

• They utilize the half-limit technique (defining $u^+$ and $u^-$ as the upper and lower semi-continuous envelopes of the limit of $u_\varepsilon$).
• They prove that $u^+$ and $u^-$ are viscosity sub- and super-solutions to the limit problem $(D_0)$.
• A comparison principle for the limit problem is established using the global ellipticity condition, ensuring $u^+ \leq u^-$, which implies convergence to a unique limit $u_0$.

3. Key Contributions

1. Relaxation of Monotonicity: The primary contribution is removing the requirement for strict monotonicity ($c_{\lambda\mu} > 0$) in the fully nonlinear operator. This allows the study of degenerate elliptic equations where the zero-order term is merely non-negative ($c_{\lambda\mu} \geq 0$).
2. Global Ellipticity Condition: The introduction of a specific global condition involving a test function $s$ to ensure solvability in the absence of strict monotonicity. This bridges the gap between degenerate ellipticity and the solvability of mixed boundary value problems.
3. Coordinate Transformation Technique: The rigorous application of a coordinate distortion to transform oblique boundary conditions into a form compatible with standard barrier constructions, even in the context of thin domains.
4. Extension of Previous Work: This work directly extends the results of [2] (which handled Dirichlet lateral conditions with strict monotonicity) to the more general and challenging non-strictly monotone case.

4. Main Results

Theorem 1 summarizes the findings:

1. Existence & Boundedness: For sufficiently small $\varepsilon > 0$, the problem $(D_\varepsilon)$ admits a bounded viscosity solution $u_\varepsilon$. Furthermore, all such solutions are uniformly bounded independent of $\varepsilon$.
2. Limit Problem: There exists a unique viscosity solution $u_0 \in C(\Omega)$ to the limit problem $(D_0)$:
   $$\begin{cases} G(D^2 u_0, Du_0, u_0, x) = 0 & \text{in } \Omega \\ u_0 = \beta(x, 0) & \text{on } \partial\Omega \end{cases}$$
   where $G$ is a fully nonlinear operator derived from the original coefficients and the geometry of the thin domain.
3. Convergence: The family of solutions converges uniformly to the limit solution:
   $$\lim_{\varepsilon \to 0^+} \sup_{u \in S_\varepsilon} \sup_{(x,y) \in \Omega_\varepsilon} |u(x,y) - u_0(x)| = 0$$
   where $S_\varepsilon$ is the set of all bounded viscosity solutions to $(D_\varepsilon)$.

5. Significance

This paper is significant for the theory of fully nonlinear PDEs and asymptotic analysis in several ways:

• Mathematical Rigor: It resolves a technical gap in the literature regarding mixed boundary conditions in thin domains when the standard strict monotonicity assumption fails. This is crucial for applications in stochastic control and game theory where coefficients may only be non-negative.
• Generality: By handling oblique conditions without strict monotonicity, the results apply to a broader class of physical and economic models where the "reaction" term is not strictly dissipative.
• Methodological Innovation: The combination of global ellipticity conditions with coordinate transformations provides a robust toolkit for analyzing degenerate elliptic problems in singularly perturbed domains.

In summary, the paper successfully establishes the well-posedness and convergence of solutions for fully nonlinear elliptic PDEs in thin domains under relaxed monotonicity assumptions, relying on a novel global ellipticity condition and a sophisticated geometric transformation of the domain.