A short note on Helmholtz decompositions for bounded domains in $\mathbb{R}^3$


  • Immanuel Anjam




In this short note we consider several widely used $\mathsf {L}^{2}$-orthogonal Helmholtz decompositions for bounded domains in $\mathbb {R}^3$. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine this global property into a local equivalent: we show that functions from these spaces have zero mean in every subdomain of specific decompositions of the domain.

An application of the zero mean properties is presented for convex domains. We introduce a specialized Poincaré-type inequality, and estimate the related unknown constant from above. The upper bound is derived using the upper bound for the Poincaré constant proven by Payne and Weinberger. This is then used to obtain a small improvement of upper bounds of two Maxwell-type constants originally proven by Pauly.

Although the two dimensional case is not considered, all derived results can be repeated in $\mathbb {R}^2$ by similar calculations.


Amrouche, C., Bernardi, C., Dauge, M., and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864. https://doi.org/10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B

Bebendorf, M., A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751–756. https://doi.org/10.4171/ZAA/1170

Filonov, N., On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, St. Petersburg Math. J. 16 (2005), no. 2, 413–416. https://doi.org/10.1090/S1061-0022-05-00857-5

Grisvard, P., Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.

Leis, R., Initial boundary value problems in mathematical physics, B. G. Teubner, Stuttgart, 1986. https://doi.org/10.1007/978-3-663-10649-4

Pauly, D., On constants in Maxwell inequalities for bounded and convex domains, J. Math. Sci. (N.Y.) 210 (2015), no. 6, 787–792. https://doi.org/10.1007/s10958-015-2590-3

Pauly, D., On Maxwell's and Poincaré's constants, Discrete Contin. Dyn. Syst. Ser. S 8 (2015), no. 3, 607–618. https://doi.org/10.3934/dcdss.2015.8.607

Pauly, D., On the Maxwell constants in 3D, Math. Methods Appl. Sci. 40 (2017), no. 2, 435–447. https://doi.org/10.1002/mma.3324

Pauly, D., Solution theory, variational formulations, and functional a posteriori error estimates for general first order systems with applications to electro-magneto-statics and more, Numer. Funct. Anal. Optim. online (2019), 97 pp. https://doi.org/10.1080/01630563.2018.1490756

Payne, L. E. and Weinberger, H. F., An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292. https://doi.org/10.1007/BF00252910

Repin, S., A posteriori estimates for partial differential equations, Radon Series on Computational and Applied Mathematics, vol. 4, Walter de Gruyter, Berlin, 2008. https://doi.org/10.1515/9783110203042

Weck, N., Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl. 46 (1974), 410–437. https://doi.org/10.1016/0022-247X(74)90250-9



How to Cite

Anjam, I. (2019). A short note on Helmholtz decompositions for bounded domains in $\mathbb{R}^3$. MATHEMATICA SCANDINAVICA, 125(2), 227–238. https://doi.org/10.7146/math.scand.a-114908