Dirichlet and Neumann problems for planar domains with parameter
HTML articles powered by AMS MathViewer
- by Florian Bertrand and Xianghong Gong PDF
- Trans. Amer. Math. Soc. 366 (2014), 159-217 Request permission
Abstract:
Let $\Gamma (\cdot ,\lambda )$ be smooth, i.e. $\mathcal C^\infty$, embeddings from $\overline {\Omega }$ onto $\overline {\Omega ^{\lambda }}$, where $\Omega$ and $\Omega ^\lambda$ are bounded domains with smooth boundary in the complex plane and $\lambda$ varies in $I=[0,1]$. Suppose that $\Gamma$ is smooth on $\overline \Omega \times I$ and $f$ is a smooth function on $\partial \Omega \times I$. Let $u(\cdot ,\lambda )$ be the harmonic functions on $\Omega ^\lambda$ with boundary values $f(\cdot ,\lambda )$. We show that $u(\Gamma (z,\lambda ),\lambda )$ is smooth on $\overline \Omega \times I$. Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings $\Gamma (\cdot ,\lambda )$ from $\overline {\mathbb D}$, the closure of the unit disc, onto $\overline {\Omega ^\lambda }$ such that $\Gamma$ is smooth on $\overline {\mathbb D}\times I$ and real analytic at $(\sqrt {-1},0)\in \overline {\mathbb D}\times I$, but for every family of Riemann mappings $R(\cdot ,\lambda )$ from $\overline {\Omega ^\lambda }$ onto $\overline {\mathbb D}$, the function $R(\Gamma (z,\lambda ),\lambda )$ is not real analytic at $(\sqrt {-1},0)\in \overline {\mathbb D}\times I$.References
- L. Bers, Riemann Surfaces (mimeographed lecture notes), New York University (1957-1958).
- F. Bertrand, X. Gong and J.-P. Rosay, Common boundary values of holomorphic functions for two-sided complex structures, submitted.
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
- L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Springer-Verlag, Berlin, 1990.
- O. D. Kellogg, Potential functions on the boundary of their regions of definition, Trans. Amer. Math. Soc. 9 (1908), no. 1, 39–50. MR 1500801, DOI 10.1090/S0002-9947-1908-1500801-0
- O. D. Kellogg, Double distributions and the Dirichlet problem, Trans. Amer. Math. Soc. 9 (1908), no. 1, 51–66. MR 1500802, DOI 10.1090/S0002-9947-1908-1500802-2
- O. D. Kellogg, Harmonic functions and Green’s integral, Trans. Amer. Math. Soc. 13 (1912), no. 1, 109–132. MR 1500909, DOI 10.1090/S0002-9947-1912-1500909-0
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317, DOI 10.1007/978-3-642-86748-4
- S.G. Mikhlin, Mathematical Physics: an advanced course, North Holland, Amsterdam, 1970.
- Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700
- Josef Plemelj, Über lineare Randwertaufgaben der Potentialtheorie, Monatsh. Math. Phys. 15 (1904), no. 1, 337–411 (German). I. Teil. MR 1547285, DOI 10.1007/BF01692306
- Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1962. MR 0150320
- S.E. Warschawski, Über einen Satz von O.D. Kellogg, Göttinger Nachrichten, Math.-Phys. Klasse, 1932, 73-86.
- Stefan Warschawski, Über das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung, Math. Z. 35 (1932), no. 1, 321–456 (German). MR 1545302, DOI 10.1007/BF01186562
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
Additional Information
- Florian Bertrand
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria
- MR Author ID: 821365
- Email: bertrand@math.wisc.edu
- Xianghong Gong
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 029815
- ORCID: 0000-0002-7065-9412
- Email: gong@math.wisc.edu
- Received by editor(s): October 31, 2011
- Published electronically: May 21, 2013
- Additional Notes: The research of the second author was supported in part by NSF grant DMS-0705426.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 159-217
- MSC (2010): Primary 31A10, 45B05, 30C35, 35B30, 32H40
- DOI: https://doi.org/10.1090/S0002-9947-2013-05951-X
- MathSciNet review: 3118395