Exact Green's formula for the fractional Laplacian and perturbations


  • Gerd Grubb




Let Ω 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 (ψdo) $P$ 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 ψdo 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.


Abatangelo, N., Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5555–5607. https://doi.org/10.3934/dcds.2015.35.5555

Abatangelo, N., Dipierro, S., Fall, M. M., Jarohs, S., and Saldaña, A., Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions, Discrete Contin. Dyn. Syst. 39 (2019), no. 3, 1205–1235. https://doi.org/10.3934/dcds.2019052

Abatangelo, N., Jarohs, S., and Saldaña, A., Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Commun. Contemp. Math. 20 (2018), no. 8, art. 1850002, 36 pp. https://doi.org/10.1142/S0219199718500025

Duduchava, L. R., Mitrea, D., and Mitrea, M., Differential operators and boundary value problems on hypersurfaces, Math. Nachr. 279 (2006), no. 9-10, 996–1023. https://doi.org/10.1002/mana.200410407

Grubb, G., Pseudo-differential boundary problems in $L_p$ spaces, Comm. Partial Differential Equations 15 (1990), no. 3, 289–340. https://doi.org/10.1080/03605309908820688

Grubb, G., Functional calculus of pseudodifferential boundary problems, second ed., Progress in Mathematics, vol. 65, Birkhäuser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-0769-6

Grubb, G., Distributions and operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009.

Grubb, G., Local and nonlocal boundary conditions for µ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE 7 (2014), no. 7, 1649–1682. https://doi.org/10.2140/apde.2014.7.1649

Grubb, G., Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. https://doi.org/10.1016/j.aim.2014.09.018

Grubb, G., Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Differential Equations 261 (2016), no. 3, 1835–1879. https://doi.org/10.1016/j.jde.2016.04.017

Grubb, G., Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr. 289 (2016), no. 7, 831–844. https://doi.org/10.1002/mana.201500041

Grubb, G., Fractional-order operators: boundary problems, heat equations, in “Mathematical analysis and applications—plenary lectures”, Springer Proc. Math. Stat., vol. 262, Springer, Cham, 2018, pp. 51–81. https://doi.org/10.1007/978-3-030-00874-1_2

Grubb, G., Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators, Comm. Partial Differential Equations 43 (2018), no. 5, 750–789. https://doi.org/10.1080/03605302.2018.1475487

Grubb, G., Limited regularity of solutions to fractional heat and Schrödinger equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 6, 3609–3634. https://doi.org/10.3934/dcds.2019148

Hörmander, L., Seminar notes on pseudo-differential operators and boundary problems, lectures at ias princeton 1965–66, Lund University Publications, working paper https://lup.lub.lu.se/search/ws/files/42232915/Lars_Hormander_Seminar_Notes_IAS_1965_66.pdf, 1966.

Hörmander, L., The analysis of linear partial differential operators. III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, 1985.

Hsiao, G. C. and Wendland, W. L., Boundary integral equations, Applied Mathematical Sciences, vol. 164, Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-68545-6

Ros-Oton, X. and Serra, J., The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 587–628. https://doi.org/10.1007/s00205-014-0740-2

Ros-Oton, X. and Serra, J., Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 2131–2150. https://doi.org/10.3934/dcds.2015.35.2131

Ros-Oton, X., Serra, J., and Valdinoci, E., Pohozaev identities for anisotropic integrodifferential operators, Comm. Partial Differential Equations 42 (2017), no. 8, 1290–1321. https://doi.org/10.1080/03605302.2017.1349148

Seeley, R. T., Complex powers of an elliptic operator, in “Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966)”, Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307.

Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

Taylor, M. E., Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981.



How to Cite

Grubb, G. (2020). Exact Green’s formula for the fractional Laplacian and perturbations. MATHEMATICA SCANDINAVICA, 126(3), 568–592. https://doi.org/10.7146/math.scand.a-120889