[Paper Review] Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains

Isabeau Birindelli, Françoise Demengel|ArXiv.org|Mar 26, 2008

Nonlinear Partial Differential Equations18 references22 citations

TL;DR

This paper establishes the existence of an eigenvalue and solutions to the Dirichlet problem for fully nonlinear, singular/degenerate elliptic operators in bounded domains satisfying only the uniform exterior cone condition. By constructing barriers via the cone condition and adapting viscosity solution techniques, it proves the existence of a positive eigenfunction and shows that the eigenvalue $λ_e$ defined via regular domains equals the supremum of $λ$ for which a positive supersolution exists, ensuring the maximum principle and solvability of the Dirichlet problem for $λ < \u03bb_e$. The key contribution is extending eigenvalue theory to non-smooth domains using geometric barrier methods instead of $C^2$ boundary regularity.

ABSTRACT

In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are similar to those of the p-Laplacian, the novelty resides in the fact that we consider the equations in bounded domains which only satisfy the exterior cone condition.

Motivation & Objective

• To extend the theory of principal eigenvalues and Dirichlet problems for fully nonlinear, singular/degenerate elliptic operators to domains that only satisfy the uniform exterior cone condition, rather than requiring $C^2$ boundaries.
• To establish the existence of a positive eigenfunction and a well-defined eigenvalue $λ_e$ in such non-smooth domains.
• To prove that $λ_e = \tilde{\lambda}$, where $\tilde{\lambda}$ is the supremum of $λ$ for which a positive supersolution exists, thereby linking the eigenvalue to a maximum principle condition.
• To demonstrate that for all $λ < \u03bb_e$, the maximum principle holds and the Dirichlet problem with negative right-hand side is solvable.

Proposed method

• Construct a barrier function $u_o$ using the uniform exterior cone condition, replacing the need for $C^2$ boundary regularity used in prior works.
• Define the eigenvalue $\bar{\lambda}(\Omega)$ as the supremum of $\lambda$ for which there exists a positive supersolution to the inequality $F[u] + h\cdot\nabla u|\nabla u|^\alpha + (V+\lambda)u^{1+\alpha} \leq 0$.
• Introduce $\lambda_e = \sup\{\lambda(\Omega') \mid \Omega \subset\subset \Omega' \text{ regular and bounded}\}$, defining the eigenvalue via approximation by smooth domains.
• Define $\tilde{\lambda}$ as the supremum of $\lambda$ for which a positive supersolution exists in $\overline{\Omega}$, and prove $\lambda_e = \tilde{\lambda}$ via compactness and viscosity solution convergence.
• Use viscosity solution theory and comparison principles to prove existence and Hölder regularity of solutions to the Dirichlet problem for $\lambda < \lambda_e$, with $f \leq 0$.
• Apply uniform Hölder estimates and barrier constructions to extract a limit solution $\phi_e > 0$ in $\Omega$ that vanishes on $\partial\Omega$, satisfying the eigenvalue equation with $\lambda_e$.

Experimental results

Research questions

• RQ1 Can the principal eigenvalue and Dirichlet problem for fully nonlinear, singular/degenerate operators be defined and solved in domains with only the uniform exterior cone condition, without $C^2$ boundary regularity?
• RQ2 Is the eigenvalue $\lambda_e$ defined via approximation by smooth domains equal to the supremum $\tilde{\lambda}$ of $\lambda$ for which a positive supersolution exists in $\overline{\Omega}$?
• RQ3 Does the maximum principle hold for $\lambda < \bar{\lambda}(\Omega)$ when the domain satisfies only the cone condition, and is this equivalent to $\lambda_e = \bar{\lambda}(\Omega)$?
• RQ4 Can the Dirichlet problem be solved for $\lambda < \lambda_e$ when the domain is non-smooth, and is the solution positive and Hölder continuous?
• RQ5 What is the relationship between the eigenvalue $\lambda_e$ and the existence of a positive eigenfunction $\phi_e$ vanishing on $\partial\Omega$?

Key findings

• The eigenvalue $\lambda_e$ defined as the supremum of $\lambda(\Omega')$ over regular domains $\Omega'$ containing $\Omega$ is equal to $\tilde{\lambda}$, the supremum of $\lambda$ for which a positive supersolution exists in $\overline{\Omega}$, proving $\lambda_e = \tilde{\lambda}$.
• There exists a positive eigenfunction $\phi_e > 0$ in $\Omega$ that satisfies the equation $F[\phi_e] + h\cdot\nabla\phi_e|\nabla\phi_e|^\alpha + (V + \lambda_e)\phi_e^{1+\alpha} = 0$ in $\Omega$ and vanishes on $\partial\Omega$, confirming the existence of a principal eigenfunction in non-smooth domains.
• For all $\lambda < \lambda_e$, the maximum principle holds, and for any continuous $f \leq 0$, the Dirichlet problem $F[u] + h\cdot\nabla u|\nabla u|^\alpha + (V + \lambda)u^{1+\alpha} = f$ in $\Omega$, $u = 0$ on $\partial\Omega$, admits a viscosity solution $u \geq 0$ that is $\gamma$-Hölder continuous.
• The solution sequence $u_n$ constructed via iteration is increasing and bounded by $Cu_o$, where $u_o$ is the barrier function, ensuring convergence to a solution via Hölder regularity and compactness.
• The value $\bar{\lambda}(\Omega)$, defined as the supremum of $\lambda$ for which a positive supersolution exists, satisfies $\lambda_e \leq \bar{\lambda}(\Omega)$, with equality if $\Omega$ is smooth, but the equality remains open for domains satisfying only the cone condition.
• The validity of the maximum principle for $\lambda < \bar{\lambda}(\Omega)$ is equivalent to $\lambda_e = \bar{\lambda}(\Omega)$ and to the solvability of the Dirichlet problem for all $\lambda < \bar{\lambda}(\Omega)$.