Exact Green's formula for the fractional Laplacian and perturbations

Abstract

Let $\Omega$ be an open, smooth, bounded subset of ${\mathbb{R}}^n$. In connection with the fractional Laplacian $(-\Delta {)}^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator ($\psi$do) $P$ with even symbol, one can define the Dirichlet value ${\gamma }_0^{a-1}u$ resp. Neumann value ${\gamma }_1^{a-1}u$ of $u(x)$ as the trace resp. normal derivative of $u/d^{a-1}$ on $\partial \Omega$, where $d(x)$ is the distance from $x\in \Omega$ to $\partial \Omega$; they define well-posed boundary value problems for $P$. A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B{\gamma }_0^{a-1}u,{\gamma }_0^{a-1}v{)}_{\partial \Omega }$, where $B$ is a first-order $\psi$do on $\partial \Omega$. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta$, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta$ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.

Title changed by insertion of the word "Exact". 22 pages. References added, misprints corrected. To appear in Mathematica Scandinavica

References in corpus (2)

• Integral representation of solutions to higher-order fractional Dirichlet problems on balls
• Limited regularity of solutions to fractional heat and Schrödinger equations

Cited by in corpus (1)

• On Landis Conjecture for the Fractional Schrödinger Equation