Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Helmholtz decomposition; Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces; variational estimate
Helmholtz decomposition; Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces; variational estimate
Summary:
We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_{p}$-weighted $L^{p}$-space $(L^{p}_{w}(\Omega ))^{n}$ for any domain $\Omega $ in $\mathbb {R}^{n}$, $n \in \mathbb {Z}$, $n\geq 2$, $1<p<\infty $ and Muckenhoupt $A_{p}$-weight $w \in A_{p}$. Set $p':={p}/{(p-1)}$ and $w':=w^{-{1}/{(p-1)}}$. Then the Helmholtz decomposition of $(L^{p}_{w}(\Omega ))^{n}$ and $(L^{p'}_{w'}(\Omega ))^{n}$ and the variational estimate of $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ are equivalent. Furthermore, we can replace $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ by $L^{p}_{w,\sigma }(\Omega )$ and $L^{p'}_{w',\sigma }(\Omega )$, respectively. The proof is based on the reflexivity and orthogonality of $L^{p}_{w,\pi }(\Omega )$ and $L^{p}_{w,\sigma }(\Omega )$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with the aid of the Helmholtz projection of $(L^{p}_{w}(\Omega ))^{n}$.
References:
[1] Rham, G. de: Variétés différentiables. Formes, courants, formes harmoniques. Publications de l'Institut de Mathématique de l'Université de Nancago III. Actualités Scientifiques et Industrielles 1222 b, Hermann, Paris (1973), French. MR 0346830 | Zbl 0284.58001
[2] Fabes, E., Mendez, O., Mitrea, M.: Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159 (1998), 323-368. DOI 10.1006/jfan.1998.3316 | MR 1658089 | Zbl 0930.35045
[3] Farwig, R.: Weighted $L^{q}$-Helmholtz decompositions in infinite cylinders and in infinite layers. Adv. Differ. Equ. 8 (2003), 357-384. MR 1948530 | Zbl 1038.35068
[4] Farwig, R., Kozono, H., Sohr, H.: An $L^{q}$-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195 (2005), 21-53. DOI 10.1007/BF02588049 | MR 2233684 | Zbl 1111.35033
[5] Farwig, R., Kozono, H., Sohr, H.: On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88 (2007), 239-248. DOI 10.1007/s00013-006-1910-8 | MR 2305602 | Zbl 1121.35097
[6] Farwig, R., Sohr, H.: Weighted $L^q$-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49 (1997), 251-288. DOI 10.2969/jmsj/04920251 | MR 1601373 | Zbl 0918.35106
[7] Fröhlich, A.: The Helmholtz decomposition of weighted $L^{q}$-spaces for Muckenhoupt weights. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46 (2000), 11-19. MR 1896920 | Zbl 1034.35089
[8] Fröhlich, A.: Maximal regularity for the non-stationary Stokes system in an aperture domain. J. Evol. Equ. 2 (2002), 471-493. DOI 10.1007/PL00012601 | MR 1941038 | Zbl 1040.35059
[9] Fröhlich, A.: The Stokes operator in weighted $L^{q}$-spaces I. Weighted estimates for the Stokes resolvent problem in a half space. J. Math. Fluid Mech. 5 (2003), 166-199. DOI 10.1007/s00021-003-0080-8 | MR 1982327 | Zbl 1040.35060
[10] Fröhlich, A.: The Stokes operator in weighted $L^{q}$-spaces II. Weighted resolvent estimates and maximal $L^{p}$-regularity. Math. Ann. 339 (2007), 287-316. DOI 10.1007/s00208-007-0114-2 | MR 2324721 | Zbl 1126.35041
[11] García-Cuerva, J., Francia, J. L. Rubio de: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116, North-Holland, Amsterdam (1985). DOI 10.1016/s0304-0208(08)x7154-3 | MR 0807149 | Zbl 0578.46046
[12] Geng, J., Shen, Z.: The Neumann problem and Helmholtz decomposition in convex domains. J. Funct. Anal. 259 (2010), 2147-2164. DOI 10.1016/j.jfa.2010.07.005 | MR 2671125 | Zbl 1195.35128
[13] Kim, A. S., Shen, Z.: The Neumann problem in $L^{p}$ on Lipschitz and convex domains. J. Funct. Anal. 255 (2008), 1817-1830. DOI 10.1016/j.jfa.2008.06.032 | MR 2442084 | Zbl 1180.35202
[14] Kobayashi, T., Kubo, T.: Weighted $L^{p}$-$L^{q}$ estimates of the Stokes semigroup in some unbounded domains. Tsukuba J. Math. 37 (2013), 179-205. DOI 10.21099/tkbjm/1389972027 | MR 3161575 | Zbl 1282.35103
[15] Lang, J., Méndez, O.: Potential techniques and regularity of boundary value problems in exterior non-smooth domains. Regularity in exterior domains. Potential Anal. 24 (2006), 385-406. DOI 10.1007/s11118-006-9008-2 | MR 2224756 | Zbl 1220.35120
[16] Maekawa, Y., Miura, H.: Remark on the Helmholtz decomposition in domains with noncompact boundary. Math. Ann. 359 (2014), 1077-1095. DOI 10.1007/s00208-014-1033-7 | MR 3231025 | Zbl 1295.35184
[17] Simader, C. G., Sohr, H., Varnhorn, W.: Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 60 (2014), 245-262. DOI 10.1007/s11565-013-0193-9 | MR 3208796 | Zbl 1304.35506
Search
Partner of
