[Paper Review] Oblique boundary value problems for augmented Hessian equations III

Feida Jiang, Neil S. Trudinger|arXiv (Cornell University)|Dec 4, 2018

Nonlinear Partial Differential Equations45 references3 citations

TL;DR

This paper establishes the existence and uniqueness of globally Lipschitz and interior $C^{1,1}$ solutions to semilinear oblique boundary value problems for degenerate and non-degenerate augmented Hessian equations on general bounded domains, without requiring domain convexity. By deriving local second derivative estimates at the boundary and proving viscosity comparison principles, the authors show that solutions achieve higher regularity near boundary points where uniform convexity conditions are satisfied.

ABSTRACT

In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for degenerate, (and non-degenerate), augmented Hessian equations, with strictly regular associated matrix functions. By establishing local second derivative estimates at the boundary and proving viscosity comparison principles, we show that the solution is correspondingly smooth near boundary points where the appropriate uniform convexity is satisfied.

Motivation & Objective

• To establish existence and uniqueness of solutions for semilinear oblique boundary value problems involving augmented Hessian equations in general bounded domains without geometric constraints.
• To prove that solutions are globally Lipschitz and possess interior $C^{1,1}$ regularity under minimal assumptions on the domain and data.
• To derive local second derivative estimates at the boundary under uniform convexity conditions, enabling higher regularity near boundary points.
• To extend the theory to degenerate cases by introducing viscosity comparison principles and refining assumptions on the operator $F$.
• To generalize prior results by removing convexity assumptions on the domain and extending to strictly regular matrix functions $A$

Proposed method

• Derive local second derivative estimates at the boundary using barrier constructions and the structure of the augmented Hessian operator $\mathcal{F}[u] = F(D^2u - A(\cdot,u,Du))$.
• Establish viscosity comparison principles for solutions in $C^{1,1}(\Omega) \cap C^{0,1}(\overline{\Omega})$ and $C^{2}(\Omega) \cap C^{0,1}(\overline{\Omega})$ under different assumptions on $F$.
• Use regularized problems with $\epsilon$-perturbations to construct approximating solutions $u_\epsilon$, then pass to the limit as $\epsilon \to 0$ to recover global regularity.
• Apply the strict subsolution method and barrier techniques from previous works to prove comparison principles in the case where $\varphi$ is strictly increasing in $z$.
• Employ the assumption that $F$ satisfies F1 (strictly increasing), F2 (concave), F3 (range condition), F4 (growth at infinity), and refined versions of F5 (uniform lower bound on directional derivative).
• Utilize orthogonally invariant structure and strict regularity of $A$ to extend results beyond the non-degenerate case, including to $a_0 = -\infty$

Experimental results

Research questions

• RQ1 Under what conditions does a solution exist for semilinear oblique boundary value problems involving degenerate augmented Hessian equations on general bounded domains?
• RQ2 How can local second derivative estimates be established at the boundary without domain convexity?
• RQ3 What conditions ensure uniqueness of $C^{1,1}$ and $C^2$ solutions in the presence of oblique boundary conditions?
• RQ4 In what cases does the viscosity solution theory extend to classical $C^{1,1}$ or $C^{2}$ solutions?
• RQ5 How does the regularity of the solution propagate from the interior to the boundary when local uniform convexity is satisfied?

Key findings

• Solutions exist and are unique in the class $C^{1,1}(\Omega) \cap C^{0,1}(\overline{\Omega})$ under the F1 - conditions, even in the degenerate case.
• Solutions are $C^{2,\alpha}(\Omega) \cap C^{0,1}(\overline{\Omega})$ under the F1 conditions, with $\alpha \in (0,1)$, in the non-degenerate case.
• Local second derivative estimates hold uniformly in a neighborhood $\mathcal{N} = B_{\theta R}(x_0) \cap \Omega$ near any boundary point $x_0 \in \partial\Omega$ where uniform $(\Gamma,A,G)$-convexity is satisfied.
• The bound $\sup_{\mathcal{N} \cap \Omega} |D^2 u_\epsilon| \leq C$ is independent of $\epsilon$, enabling the limiting process $\epsilon \to 0$ to recover $C^{1,1}$ regularity.
• Uniqueness in $C^{1,1}(\Omega) \cap C^{0,1}(\overline{\Omega})$ follows from the comparison principle in case (i) when $\varphi$ is strictly increasing in $z$, and in case (ii) when $A$ is strictly increasing in $z$.
• The results extend to the case $a_0 = -\infty$ by redefining the domain $D = \{ r \in \Gamma \mid F(r) > a_0 \}$, preserving existence and regularity under the same structural assumptions.