[Paper Review] ${\cal C}^{1,\beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations

Isabeau Birindelli, F. Demengel|arXiv (Cornell University)|Aug 2, 2012

Nonlinear Partial Differential Equations7 references4 citations

TL;DR

This paper establishes $C^{1,eta}$ regularity for viscosity solutions of fully nonlinear degenerate elliptic equations of the form $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) = f$ in bounded $C^2$ domains, extending prior interior regularity results to the boundary. The authors introduce new boundary flatness improvement techniques and handle the degeneracy via weighted barrier functions and compactness arguments, proving that solutions are Hölder continuous in the gradient up to the boundary when $\alpha \geq 0$, with explicit Hölder exponent $\beta$ depending on $\alpha$, $N$, and ellipticity constants.

ABSTRACT

In this paper we prove Holder regularity of the gradient for solutions of Dirichlet problem associate to degenerate elliptic equations, extending the recent result of Imbert and Silvestre. Indeed we obtain regularity up to the boundary and when the equation has lower order terms.The proof follows their scheme but requires new tools and new ideas. In particular we give some a priori Lipschitz and H\"older estimates in the presence of boundary condition on one part of the boundary.

Motivation & Objective

• To establish $C^{1,\beta}$ regularity up to the boundary for viscosity solutions of degenerate fully nonlinear elliptic equations with $\alpha \geq 0$, extending prior interior regularity results.
• To resolve the open problem of gradient regularity near the boundary for equations of the form $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) = f$, which is essential for proving simplicity of generalized principal eigenvalues.
• To develop new boundary flatness improvement techniques and compactness arguments that handle the degeneracy introduced by $|\nabla u|^\alpha$ and the lower-order term $h(x) \cdot \nabla u$.
• To prove that the generalized principal eigenvalue for the operator $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) + \bar{\lambda} u^{1+\alpha} = 0$ is simple in $C^2$ domains with connected boundary.
• To extend the regularity theory to include lower-order terms and boundary conditions, providing a complete $C^{1,\beta}$ estimate with explicit dependence on data.

Proposed method

• Uses an improvement of flatness lemma adapted to degenerate equations with $|\nabla u|^\alpha$ and boundary conditions, proving that oscillation of $u - p \cdot x$ decays like $r^{1+\beta}$.
• Applies a comparison principle for viscosity solutions with lower-order terms $h(x) \cdot \nabla u$, ensuring comparison even in degenerate regimes.
• Employs weighted barrier functions and distance-to-boundary estimates to control boundary behavior, especially when $\nabla u$ is small.
• Uses compactness arguments: bounded sequences of solutions converge uniformly on compact subsets, and the limit equation inherits the structure of the original.
• Applies a blow-up argument and contradiction to show that the gradient remains Hölder continuous up to the boundary, relying on the boundedness of gradient sequences.
• Introduces a technical proposition proving that sequences $p_n$ satisfying $|p_n \cdot (x', a(x'))| \leq C$ are bounded when $a$ is $C^2$, $a(0) = \nabla a(0) = 0$, and $a$ not identically zero.

Experimental results

Research questions

• RQ1 Does $C^{1,\beta}$ regularity hold for viscosity solutions of $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) = f$ up to the boundary when $\alpha \geq 0$?
• RQ2 Can the strong comparison principle be applied near the boundary for such equations, given the degeneracy when $|\nabla u| \to 0$?
• RQ3 Is the generalized principal eigenvalue for the operator $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) + \bar{\lambda} u^{1+\alpha} = 0$ simple in $C^2$ domains with connected boundary?
• RQ4 How does the addition of a lower-order term $h(x) \cdot \nabla u$ affect the regularity and compactness of solutions near the boundary?
• RQ5 What is the optimal Hölder exponent $\beta$ for the gradient in terms of $\alpha$, $N$, and ellipticity constants?

Key findings

• The paper establishes a global $C^{1,\beta}$ estimate for viscosity solutions: $\|u\|_{C^{1,\beta}(\Omega)} \leq C\left(\|\phi\|_{C^{1,\beta_0}(\partial\Omega)} + \|u\|_{L^\infty(\Omega)} + \|f\|_{L^\infty(\Omega)}^{1/(1+\alpha)}\right)$, with $\beta$ depending on $\lambda, \Lambda, \|f\|_\infty, N, \Omega, \|h\|_\infty, \beta_0$.
• The Hölder exponent $\beta$ is explicitly quantified and depends on the ellipticity constants $\lambda, \Lambda$, the dimension $N$, and the data norms.
• The authors prove that the generalized principal eigenvalue $\bar{\lambda}$ for the operator $|\nabla u|^\alpha(F(D^2u) + h(x) \cdot \nabla u) + \bar{\lambda} u^{1+\alpha} = 0$ is simple, meaning any two positive eigenfunctions are scalar multiples of each other.
• The proof relies on the $C^{1,\beta}$ regularity up to the boundary, which ensures the strong comparison principle holds near the boundary, enabling the simplicity proof.
• The method introduces a new boundary flatness improvement lemma that accounts for the degeneracy and the lower-order term, extending the interior argument of Imbert and Silvestre to the boundary.
• A key technical result shows that sequences $p_n$ satisfying $|p_n \cdot (x', a(x'))| \leq C$ are bounded when $a \in C^2$, $a(0) = \nabla a(0) = 0$, and $a$ not identically zero, which is essential for compactness.