[Paper Review] Smooth Solutions to a Class of Nonlocal Fully Nonlinear Elliptic Equations

Hui Yu|arXiv (Cornell University)|Nov 13, 2015

Nonlinear Partial Differential Equations8 references3 citations

TL;DR

This paper establishes the existence of smooth solutions to a broad class of nonlocal fully nonlinear elliptic equations by developing a robust bootstrap regularity theory. It proves that solutions are $C^inity$ in the interior of the domain when the right-hand side is smooth and the boundary data are bounded and uniformly continuous, overcoming the challenge of global dependence in nonlocal operators through a localized, iterative regularity improvement strategy.

ABSTRACT

We show that a certain class of fully nonlinear nonlocal equations have smooth solutions as long as the right-hand side is nice and the boundary datum is bounded. To this end we follow the classical strategy. We first show that solutions are $C^{σ+α}$ continuous, then develop a bootstrap argument robust enough for operators that may depend on the solution and its σ-derivatives.

Motivation & Objective

• To establish a general framework for proving smooth solutions to nonlocal fully nonlinear elliptic equations.
• To address the challenge of global dependence in nonlocal operators, which hinders classical regularity methods.
• To extend the bootstrap argument to nonlocal equations where coefficients depend on the solution and its nonlocal derivatives.
• To show that smoothness of the right-hand side and boundedness of boundary data suffice for $C^\infty$ interior regularity.
• To provide applications to linear and nonlinear operators, including those based on eigenvalues of the nonlocal Hessian $D^\sigma_\rho u$.

Proposed method

• Introduces a nonlocal Hessian operator $D^\sigma_\rho u$ defined via a weight $\rho$ and directional differences $\delta u(x,y)$, generalizing the second-order Hessian.
• Establishes a $C^{\sigma+\alpha}$ regularity estimate for viscosity solutions using a perturbative argument and comparison principle.
• Develops a localized bootstrap procedure that iteratively improves regularity by differentiating the equation and applying Schauder-type estimates.
• Uses a cutoff function technique to localize the nonlocal operator and control boundary effects, enabling iterative regularity gain.
• Applies the theory to directional derivatives, showing that each derivative satisfies an equation of the same type with Hölder continuous coefficients.
• Employs a localization-bootstrap iteration to upgrade regularity from $C^{\sigma+\alpha}$ to $C^{2+\sigma+\alpha}$, and so on, to $C^\infty$.

Experimental results

Research questions

• RQ1 Can smooth solutions be established for a general class of nonlocal fully nonlinear elliptic equations despite global dependence in the operator?
• RQ2 Does the bootstrap method used in second-order PDEs extend to nonlocal equations with solution-dependent coefficients?
• RQ3 Can the regularity of solutions be improved iteratively when the nonlocal operator depends on the solution globally?
• RQ4 What conditions on the right-hand side and boundary data are sufficient for $C^\infty$ regularity in the interior?
• RQ5 Can the theory be applied to operators defined via eigenvalues of the nonlocal Hessian $D^\sigma_\rho u$?

Key findings

• The Dirichlet problem for $F(D^\sigma_\rho u, x) = f(x)$ in $B_1$ admits a unique smooth solution in the interior if $f$ is smooth and $g$ is bounded and uniformly continuous.
• Solutions are shown to be $C^\infty$ in $B_1$ via an iterative localization-bootstrap argument, even when the operator depends globally on $u$.
• The $C^{\sigma+\alpha}$ regularity estimate is established as the foundational step, enabling the bootstrap process.
• For linear operators of the form $\int \delta u(x,y) \frac{\langle A(x)y,y\rangle \rho(x,y)}{|y|^{n+\sigma+2}} dy = f(x)$, smooth solutions exist under the same conditions.
• The theory applies to operators based on eigenvalues of $D^\sigma_\rho u$, such as $F(M) = f(\lambda_1, \dots, \lambda_n)$, with $f$ smooth and concave.
• Even for less regular operators, the method yields $C^{k-1+\sigma+\alpha}$ regularity if $f$ is $C^k$ and $g$ is bounded, provided a priori $C^k$ bounds on $u$ are available.