A generalization of Dahlberg’s theorem concerning the regularity of harmonic Green potentials
HTML articles powered by AMS MathViewer
- by Dorina Mitrea PDF
- Trans. Amer. Math. Soc. 360 (2008), 3771-3793 Request permission
Abstract:
Let $\mathbb {G}_D$ be the solution operator for $\Delta u = f$ in $\Omega$, Tr $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb {R}^n$. B. E. J. Dahlberg proved that for a bounded Lipschitz domain $\Omega , \nabla \mathbb {G}_D$ maps $L^1 (\Omega )$ boundedly into weak-$L^1(\Omega )$ and that there exists $p_n > 1$ such that $\nabla \mathbb {G}_D : L^p (\Omega )\rightarrow L^{p^{*}} (\Omega )$ is bounded for $1 < p < n, \frac {1}{p^*} = \frac {1}{p} - \frac {1}{n}$. In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator $\mathbb {G}_N$ corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
- Russell M. Brown, The Neumann problem on Lipschitz domains in Hardy spaces of order less than one, Pacific J. Math. 171 (1995), no. 2, 389–407. MR 1372235
- A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
- Björn E. J. Dahlberg, $L^{q}$-estimates for Green potentials in Lipschitz domains, Math. Scand. 44 (1979), no. 1, 149–170. MR 544584, DOI 10.7146/math.scand.a-11800
- Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI 10.2307/1971407
- Eugene Fabes, Osvaldo Mendez, and Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), no. 2, 323–368. MR 1658089, DOI 10.1006/jfan.1998.3316
- C. Fefferman, N. M. Rivière, and Y. Sagher, Interpolation between $H^{p}$ spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75–81. MR 388072, DOI 10.1090/S0002-9947-1974-0388072-3
- John E. Gilbert and Margaret A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, vol. 26, Cambridge University Press, Cambridge, 1991. MR 1130821, DOI 10.1017/CBO9780511611582
- David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
- Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
- David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
- Svitlana Mayboroda and Marius Mitrea, Sharp estimates for Green potentials on non-smooth domains, Math. Res. Lett. 11 (2004), no. 4, 481–492. MR 2092902, DOI 10.4310/MRL.2004.v11.n4.a7
- Osvaldo Mendez and Marius Mitrea, Complex powers of the Neumann Laplacian in Lipschitz domains, Math. Nachr. 223 (2001), 77–88. MR 1817850, DOI 10.1002/1522-2616(200103)223:1<77::AID-MANA77>3.3.CO;2-4
- Dorina Mitrea, On the regularity of the harmonic Green potential in nonsmooth domains, Integral methods in science and engineering, Birkhäuser Boston, Boston, MA, 2006, pp. 177–188. MR 2181936, DOI 10.1007/0-8176-4450-4_{1}5
- Dorina Mitrea, Layer potentials and Hodge decompositions in two dimensional Lipschitz domains, Math. Ann. 322 (2002), no. 1, 75–101. MR 1883390, DOI 10.1007/s002080100266
- Marius Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Mathematics, vol. 1575, Springer-Verlag, Berlin, 1994. MR 1295843, DOI 10.1007/BFb0073556
- Kaj Nyström, Integrability of Green potentials in fractal domains, Ark. Mat. 34 (1996), no. 2, 335–381. MR 1416671, DOI 10.1007/BF02559551
- Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319, DOI 10.1515/9783110812411
- Vyacheslav S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2) 60 (1999), no. 1, 237–257. MR 1721827, DOI 10.1112/S0024610799007723
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Hans Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15 (2002), no. 2, 475–524. MR 1951822, DOI 10.5209/rev_{R}EMA.2002.v15.n2.16910
- Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
Additional Information
- Dorina Mitrea
- Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
- MR Author ID: 344702
- ORCID: 0000-0002-0051-7048
- Received by editor(s): May 22, 2006
- Published electronically: February 27, 2008
- Additional Notes: The author was supported in part by NSF FRG Grant #0456306
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3771-3793
- MSC (2000): Primary 35J05, 46E35; Secondary 42B20, 34B27
- DOI: https://doi.org/10.1090/S0002-9947-08-04384-5
- MathSciNet review: 2386245