[Paper Review] The Dirichlet elliptic problem involving regional fractional Laplacian

Huyuan Chen|arXiv (Cornell University)|Sep 19, 2015

Nonlinear Partial Differential Equations10 references3 citations

TL;DR

This paper establishes the existence and uniqueness of weak and very weak solutions for the Dirichlet problem involving the regional fractional Laplacian $(-\Delta)^\alpha_\Omega$ in a bounded $C^2$ domain $\Omega \subset \mathbb{R}^N$ with $\alpha \in (1/2, 1)$. It proves that solutions exist for data $f \in L^2(\Omega)$, $f \in L^1(\Omega, \rho^\beta dx)$, and $f \in \mathcal{M}(\Omega, \rho^\beta)$, where $\rho(x) = \text{dist}(x, \partial\Omega)$ and $\beta = 2\alpha - 1$, with sharp $L^1$ and $H^\alpha_0$ estimates.

ABSTRACT

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^α_Ωu=f \qquad & { m in}\quad Ω,\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=g\quad & { m on}\quad \partial Ω, \end{array} \end{equation} where $Ω$ is a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partialΩ$, $α\in(\frac12,1)$ and the operator $(-Δ)^α_Ω$ denotes the regional fractional Laplacian. We prove that when $g\equiv0$, problem ( ef{0}) admits a unique weak solution in the cases that $f\in L^2(Ω)$, $f\in L^1(Ω, ρ^βdx)$ and $f\in \mathcal{M}(Ω,ρ^β)$, here $ρ(x)={ m dist}(x,\partialΩ)$, $β=2α-1$ and $\mathcal{M}(Ω,ρ^β)$ is a space of all Radon measures $ν$ satisfying $\int_Ωρ^βd|ν|

Motivation & Objective

• To establish the existence and uniqueness of weak solutions for the regional fractional Laplacian Dirichlet problem with zero boundary data.
• To extend the solvability framework to data in $L^1(\Omega, \rho^\beta dx)$ and Radon measures in $\mathcal{M}(\Omega, \rho^\beta)$, where $\beta = 2\alpha - 1$.
• To derive an integral by parts formula for classical solutions with general boundary data $g \in C^2(\partial\Omega)$.
• To characterize the appropriate test function spaces $\mathbb{X}_\alpha$ and $\mathbb{D}_\beta$ for very weak solutions and boundary terms.

Proposed method

• Define the regional fractional Laplacian $(-\Delta)^\alpha_\Omega$ as a limit of truncated singular integrals over $\Omega$, excluding the singularity at $x$.
• Use the Hilbert space $H^\alpha_0(\Omega)$ with Gagliardo-type norm and scalar product to define weak solutions via variational formulation.
• Introduce the space $\mathbb{X}_\alpha$ of continuous functions vanishing on $\partial\Omega$ with $\|(-\Delta)^\alpha_\Omega \xi\|_{L^\infty(\Omega)} < \infty$, and $\mathcal{M}(\Omega, \rho^\beta)$ for measure data.
• Define very weak solutions via duality: $\int_\Omega u (-\Delta)^\alpha_\Omega \xi \, dx = \int_\Omega \xi \, df$ for all $\xi \in \mathbb{X}_\alpha$.
• Apply the Dunford-Pettis theorem to extract weak limits in $L^1(\Omega)$ from approximating sequences.
• Use the integral by parts formula $\int_\Omega u (-\Delta)^\alpha_\Omega v \, dx = \int_\Omega v (-\Delta)^\alpha_\Omega u \, dx + \int_{\partial\Omega} v \frac{\partial^\beta u}{\partial \vec{n}^\beta} d\omega - \int_{\partial\Omega} u \frac{\partial^\beta v}{\partial \vec{n}^\beta} d\omega$ for $u,v \in \mathbb{D}_\beta$.
• Construct a harmonic extension $G$ of boundary data $g$ and decompose the solution as $u_g = u_0 + G$, using regularity of $G$ and convergence in weighted $L^\infty$ norms.

Experimental results

Research questions

• RQ1 Under what conditions on $f$ does the regional fractional Laplacian Dirichlet problem admit a unique weak solution in $H^\alpha_0(\Omega)$?
• RQ2 Can the solvability framework be extended to $f \in L^1(\Omega, \rho^\beta dx)$ and $f \in \mathcal{M}(\Omega, \rho^\beta)$ with $\beta = 2\alpha - 1$?
• RQ3 What is the appropriate test function space for very weak solutions when $f$ is a measure?
• RQ4 How can an integral by parts formula be derived for classical solutions with non-zero boundary data?
• RQ5 Does the solution $u_{f,g}$ of the non-homogeneous problem belong to the space $\mathbb{D}_\beta$ under regularity assumptions on $f$ and $g$?

Key findings

• For $f \in L^2(\Omega)$, the problem admits a unique weak solution $u_f \in H^\alpha_0(\Omega)$ satisfying $\|u_f\|_{H^\alpha_0(\Omega)} \leq c_1 \|f\|_{L^2(\Omega)}$.
• For $f \in \mathcal{M}(\Omega, \rho^\beta)$, the problem admits a unique very weak solution $u_f \in L^1(\Omega)$ with $\|u_f\|_{L^1(\Omega)} \leq c_2 \|f\|_{\mathcal{M}(\Omega, \rho^\beta)}$.
• The solution sequence $\{u_n\}$ is uniformly bounded and uniformly integrable in $L^1(\Omega)$, ensuring weak compactness via the Dunford-Pettis theorem.
• The integral by parts formula holds for $u,v \in \mathbb{D}_\beta$: $\int_\Omega u (-\Delta)^\alpha_\Omega v \, dx = \int_\Omega v (-\Delta)^\alpha_\Omega u \, dx + \int_{\partial\Omega} v \frac{\partial^\beta u}{\partial \vec{n}^\beta} d\omega - \int_{\partial\Omega} u \frac{\partial^\beta v}{\partial \vec{n}^\beta} d\omega$.
• For $f \in C^2(\Omega) \cap C(\bar{\Omega})$ and $g \in C^2(\partial\Omega)$, the classical solution satisfies $\int_\Omega u (-\Delta)^\alpha_\Omega v \, dx = \int_\Omega f v \, dx + \int_{\partial\Omega} g \frac{\partial^\beta v}{\partial \vec{n}^\beta} d\omega$ for all $v \in \mathbb{X}_\alpha \cap \mathbb{D}_\beta$.
• The function $G$ extending $g$ to $\bar{\Omega}$ satisfies $|(-\Delta)^\alpha_\Omega G(x)| \leq c_{55} \rho(x)^{-\beta}$, ensuring control in weighted $L^\infty$ norms.