[Paper Review] Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Xavier Ros‐Oton, Joaquim Serra|arXiv (Cornell University)|Jun 4, 2014

Nonlinear Partial Differential Equations24 references4 citations

TL;DR

This paper establishes a local integration by parts formula and a Pohozaev identity for the higher-order fractional Laplacian $(-\Delta)^s$ with $s > 1$, extending prior results for $s \in (0,1)$. The key contribution is the derivation of boundary terms involving $u/d^s$, enabling a unique continuation property for eigenfunctions of $(-\Delta)^s$ in bounded domains with zero exterior data.

ABSTRACT

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-Δ)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$. As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-Δ)^sϕ=λϕ$ in $Ω$, $ϕ\equiv0$ in $\mathbb R^n\setminusΩ$.

Motivation & Objective

• To extend the integration by parts and Pohozaev identities from the case $s \in (0,1)$ to higher-order fractional Laplacians with $s > 1$.
• To establish a local boundary term formulation for $(-\Delta)^s$ that remains valid in bounded smooth domains.
• To prove a unique continuation property for eigenfunctions of $(-\Delta)^s$ satisfying $(-\Delta)^s\phi = \lambda\phi$ in $\Omega$ and $\phi \equiv 0$ in $\mathbb{R}^n \setminus \Omega$.
• To provide a rigorous analytical framework for higher-order fractional PDEs using the regularity results of Grubb [13] and the function $d(x)$ as a boundary-weighted smoothing of distance.

Proposed method

• Derive an integration by parts formula for $(-\Delta)^s$ with $s > 1$ using the extension of the $s \in (0,1)$ results from [29], leveraging the function $d(x)$ to define a smooth boundary weight.
• Use the regularity theory of Grubb [13] to establish that $u/d^s \in C^\infty(\overline{\Omega})$ for solutions $u$ to $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\mathbb{R}^n \setminus \Omega$, under $g \in L^\infty(\Omega)$.
• Apply a density argument using smooth approximations $g_k \to g$ to extend the identity from smooth to $L^\infty$ data, ensuring convergence of $u_k/d^s$ to $u/d^s$ in $C^\alpha(\overline{\Omega})$.
• Compute the constant $\Gamma(1+s)^2$ in the Pohozaev identity via explicit computation of $(-\Delta)^s(1 - |x|^2)_+^s$ using hypergeometric functions and the fractional Laplacian's action on monomials.
• Use the Pohozaev identity to derive a unique continuation result by showing that the boundary integral term vanishes under eigenfunction conditions, forcing $\phi \equiv 0$.

Experimental results

Research questions

• RQ1 Can a local integration by parts formula be established for the higher-order fractional Laplacian $(-\Delta)^s$ with $s > 1$, analogous to the $s \in (0,1)$ case?
• RQ2 What is the precise form of the boundary term in the Pohozaev identity for $(-\Delta)^s$ with $s > 1$, and how does it depend on $u/d^s$?
• RQ3 Does the Pohozaev identity for $(-\Delta)^s$ with $s > 1$ imply a unique continuation property for eigenfunctions vanishing outside $\Omega$?
• RQ4 How can the constant $\Gamma(1+s)^2$ in the Pohozaev identity be explicitly computed for non-integer $s > 1$?
• RQ5 To what extent do the regularity results of Grubb [13] enable the derivation of local identities involving $u/d^s$ for $s > 1$?

Key findings

• An integration by parts formula is established for $(-\Delta)^s$ with $s > 1$, involving a local boundary term proportional to $\Gamma(1+s)^2 \int_{\partial\Omega} \left(\frac{u}{d^s}\right)^2 (x \cdot \nu) \, d\sigma$, valid for $u \in H^s(\mathbb{R}^n)$ with $(-\Delta)^s u \in L^\infty(\Omega)$.
• The Pohozaev identity for $(-\Delta)^s$ with $s > 1$ is derived in the form $\int_\Omega (x \cdot \nabla u)(-\Delta)^s u \, dx = \frac{2s - n}{2} \int_\Omega u (-\Delta)^s u \, dx - \Gamma(1+s)^2 \int_{\partial\Omega} \left(\frac{u}{d^s}\right)^2 (x \cdot \nu) \, d\sigma$, with the boundary term explicitly identified.
• The constant $\Gamma(1+s)^2$ in the Pohozaev identity is rigorously computed via the action of $(-\Delta)^s$ on the function $(1 - |x|^2)_+^s$, using hypergeometric function identities and the fractional Laplacian's behavior on monomials.
• A unique continuation property is proven: if $\phi \in H^s(\mathbb{R}^n)$ solves $(-\Delta)^s \phi = \lambda \phi$ in $\Omega$ and $\phi \equiv 0$ in $\mathbb{R}^n \setminus \Omega$, then $\phi \equiv 0$ in $\Omega$, due to the vanishing of the boundary integral term in the Pohozaev identity.
• The regularity result $u/d^s \in C^\infty(\overline{\Omega})$ for solutions to $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\mathbb{R}^n \setminus \Omega$, with $g \in C^\alpha(\overline{\Omega})$, is used to justify the pointwise convergence of approximating sequences in the density argument.