A Lions' type formula for some reproducing kernel Hilbert spaces of fractional harmonic functions
Sidy M. Djitte, Franck Sueur
TL;DR
The paper generalizes Lions' boundary-representation framework from classical harmonic functions to fractional harmonic functions governed by the fractional Laplacian $(-\Delta)^a$ for $a\in(0,1)$. It develops $a$-transmission Sobolev spaces and associated $a$-harmonic trace spaces, establishing a fractional Poisson representation and proving that the corresponding spaces $\mathcal{H}_{a,s}(\Omega)$ are RKHS with kernels given by boundary integrals of the fractional Green function $G_a$ and its traces. A key result is the explicit fractional Lions-type kernel $K_{a,s}(x,y)$, which recovers the classical Lions kernel in the limit $a\to1^-$ and aligns with a Hadamard-type variation formula for $G_a$ in the fractional setting. Additionally, the paper discusses RKHS for the steady Stokes problem, illustrating both similarities and distinctions with Hadamard variations, thereby enriching boundary-representation theory for a broad class of elliptic systems.
Abstract
In \cite{Lions}, J. L. Lions considered a reproducing kernel Hilbert space (RKHS) of harmonic functions on a regular domain with Sobolev traces and obtained a formula that expresses the kernel of this space as an integral on the boundary of some derivatives of the Green function associated with the Laplace operator and the homogeneous Dirichlet boundary condition. This result was simplified and extended later by Englis, Lukkassen, Peetre, and Persson in \cite{ELPL} to more general elliptic systems of even orders. In particular, they emphasized that the resemblance between Lions' type formula and the Hadamard variational formula only appears when the operator is of order $2$. In this paper, we investigate some RKHS of $a$-harmonic functions, where $a$ in $(0,1)$ refers to a fractional exponent of the Laplace operator. For such fractional order pseudo-differential operators, the local nonhomogeneous Dirichlet problem can be addressed by means of some $a$-transmission Sobolev spaces, which were introduced by Hörmander in the sixties and recently developed by Grubb in a series of papers. We deduce from these works a fractional Poisson formula, which is applied to obtain a Lions' type formula. We observe, in particular, that despite the order of the operator not being $2$, this formula resembles the Hadamard variational formula that we prove in the companion paper \cite{sidy-franck_1}. As a complementary remark, we observe that for a family of RKHS associated with the steady Stokes system, a second order system, there is also a Lions' type formula for their two-point kernels, which turn out not to be similar to the corresponding Hadamard variation formula.