Bounds and representations of solutions of planar div-curl problems
Author:
Giles Auchmuty
Journal:
Quart. Appl. Math. 75 (2017), 505-524
MSC (2010):
Primary 35J05; Secondary 35J56, 35Q60, 35C99
DOI:
https://doi.org/10.1090/qam/1463
Published electronically:
March 15, 2017
MathSciNet review:
3636166
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Abstract: Estimates and representations of solutions of div-curl systems for planar vector fields are described. Potentials are used to represent solutions as the sum of fields that depend on the source terms and harmonic fields dependent on the boundary data. Sharp 2-norm (energy) bounds for the least energy solutions on bounded regions with Lipschitz boundary are found. Prescribed flux, tangential or mixed flux and tangential boundary conditions require different potentials. The harmonic fields are represented and estimated using Steklov eigenfunctions. Some regularity results are obtained.
References
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- Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille, Variational analysis in Sobolev and BV spaces, MPS/SIAM Series on Optimization, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. Applications to PDEs and optimization. MR 2192832
- Giles Auchmuty, Orthogonal decompositions and bases for three-dimensional vector fields, Numer. Funct. Anal. Optim. 15 (1994), no. 5-6, 455–488. MR 1281557, DOI https://doi.org/10.1080/01630569408816576
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- Giles Auchmuty, Spectral characterization of the trace spaces $H^s(\partial \Omega )$, SIAM J. Math. Anal. 38 (2006), no. 3, 894–905. MR 2262947, DOI https://doi.org/10.1137/050626053
- Giles Auchmuty, Reproducing kernels for Hilbert spaces of real harmonic functions, SIAM J. Math. Anal. 41 (2009), no. 5, 1994–2009. MR 2578795, DOI https://doi.org/10.1137/080739628
- Giles Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012), no. 1, 394–406. MR 2885782, DOI https://doi.org/10.1016/j.jmaa.2012.01.051
- G. Auchmuty, The SVD of the Poisson Kernel, J. Fourier Analysis and Applications (2016). doi:10.1007/s00041-016-9515-5
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- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- K. O. Friedrichs, Differential forms on Riemannian manifolds, Comm. Pure Appl. Math. 8 (1955), 551–590. MR 87763, DOI https://doi.org/10.1002/cpa.3160080408
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Michal Křížek and Pekka Neittaanmäki, On the validity of Friedrichs’ inequalities, Math. Scand. 54 (1984), no. 1, 17–26. MR 753060, DOI https://doi.org/10.7146/math.scand.a-12037
- Dorina Mitrea, Integral equation methods for div-curl problems for planar vector fields in nonsmooth domains, Differential Integral Equations 18 (2005), no. 9, 1039–1054. MR 2162986
- Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447
- Hermann Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444. MR 3331
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497
References
- Giles Auchmuty and James C. Alexander, $L^2$ well-posedness of planar div-curl systems, Arch. Ration. Mech. Anal. 160 (2001), no. 2, 91–134. MR 1864837, DOI https://doi.org/10.1007/s002050100156
- Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille, Variational analysis in Sobolev and BV spaces, MPS/SIAM Series on Optimization, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. Applications to PDEs and optimization. MR 2192832
- Giles Auchmuty, Orthogonal decompositions and bases for three-dimensional vector fields, Numer. Funct. Anal. Optim. 15 (1994), no. 5-6, 455–488. MR 1281557, DOI https://doi.org/10.1080/01630569408816576
- Giles Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004), no. 3-4, 321–348. MR 2072072, DOI https://doi.org/10.1081/NFA-120039655
- Giles Auchmuty, Spectral characterization of the trace spaces $H^s(\partial \Omega )$, SIAM J. Math. Anal. 38 (2006), no. 3, 894–905. MR 2262947, DOI https://doi.org/10.1137/050626053
- Giles Auchmuty, Reproducing kernels for Hilbert spaces of real harmonic functions, SIAM J. Math. Anal. 41 (2009), no. 5, 1994–2009. MR 2578795, DOI https://doi.org/10.1137/080739628
- Giles Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012), no. 1, 394–406. MR 2885782, DOI https://doi.org/10.1016/j.jmaa.2012.01.051
- G. Auchmuty, The SVD of the Poisson Kernel, J. Fourier Analysis and Applications (2016). doi:10.1007/s00041-016-9515-5
- Jürgen Bolik and Wolf von Wahl, Estimating $\nabla \textbf {u}$ in terms of $\textrm {div} \textbf {u}$, $\textrm {curl} \textbf {u}$, either $(\nu ,\textbf {u})$ or $\nu \times \textbf {u}$ and the topology, Math. Methods Appl. Sci. 20 (1997), no. 9, 737–744. MR 1446207, DOI https://doi.org/10.1002/%28SICI%291099-1476%28199706%2920%3A9%24%5Clangle%24737%3A%3AAID-MMA863%24%5Crangle%243.3.CO%3B2-9
- James H. Bramble and Joseph E. Pasciak, A new approximation technique for div-curl systems, Math. Comp. 73 (2004), no. 248, 1739–1762. MR 2059734, DOI https://doi.org/10.1090/S0025-5718-03-01616-8
- Jean Bourgain and Haïm Brezis, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 277–315. MR 2293957, DOI https://doi.org/10.4171/JEMS/80
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367
- Emmanuele DiBenedetto, Real analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1897317
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- K. O. Friedrichs, Differential forms on Riemannian manifolds, Comm. Pure Appl. Math. 8 (1955), 551–590. MR 0087763, DOI https://doi.org/10.1002/cpa.3160080408
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Michal Křížek and Pekka Neittaanmäki, On the validity of Friedrichs’ inequalities, Math. Scand. 54 (1984), no. 1, 17–26. MR 753060, DOI https://doi.org/10.7146/math.scand.a-12037
- Dorina Mitrea, Integral equation methods for div-curl problems for planar vector fields in nonsmooth domains, Differential Integral Equations 18 (2005), no. 9, 1039–1054. MR 2162986
- Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447
- Hermann Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444. MR 0003331
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497
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Additional Information
Giles Auchmuty
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3008
MR Author ID:
28195
Email:
auchmuty@uh.edu
Keywords:
Div-curl,
planar vector fields,
stream function,
potential representations,
harmonic fields
Received by editor(s):
January 3, 2017
Published electronically:
March 15, 2017
Additional Notes:
The author gratefully acknowledges research support by NSF award DMS 11008754
Article copyright:
© Copyright 2017
Brown University
