Recent Scientific Publications: Giles Auchmuty:

The publications below are arranged by subject area.  For reprints or further information, send e-mail to auchmuty@uh.edu

A.     Representations of Solutions of Elliptic Boundary Value Problems.  

         Most of the following papers treat issues related to the dependence of solutions of boundary value problems on boundary data. Many use specific orthogonal bases of boundary trace  spaces  - based on the results  described in

  1.         "Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems", Num. Functional Analysis and Optimization, 25, (2004), 321-348. 
  2.         "Spectral Characterization of the Trace Spaces H^s(bdy)", SIAM J of Math. Anal., 38, (2006) 894-905. 
  3.         "Reproducing Kernels and Hilbert Spaces of Real Harmonic Functions",  SIAM J Math Analysis, 41 (2009) 1994-2009.

         Recent papers (since 2009)  include the following
        4.       "The S.V.D. of the Poisson Kernel", J. Fourier Analysis and Applications, November 2016, pp 1-20 .  http://rdcu.be/mEqr   DOI: 10.1007/s00041-016-9515-5
        5.       "Steklov Representations of Green's Functions for Laplacian Boundary Value Problems", Applied Mathematics and Optimization (to appear; online 25.7.2016).  DOI: 10.1007/s00245-016-9370-4 pp 1-23. [.pdf]
        6. (with Q. Han), ""Well-posedness and approximation of solutions of linear divergence-form elliptic problems on exterior regions", Math Methods in the Applied Sciences, 38, (2015), 1867-1875.

  1.   (with Qi Han), "Spectral Representation of Solutions of Linear Elliptic Equations on Exterior Regions", J. Math. Anal & Appns, 361, (2013), 1-10.
  2.   (with Qi Han), "Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions", Applied Math & Optimization, 69 (2014), 21-45.
  3.   (with P. Kloucek), "Spectral solutions of Self-adjoint Elliptic Problems with Immersed Interfaces", Applied Mathematics and Optimization, 64. (2011), 311-338.
  4.   "Finite energy solutions of Self-adjoint Elliptic Mixed Boundary Value Problems",  Mathematical Methods in the Applied Sciences. 33, (2010) 1446-146.



B.     Analysis of Vector Fields and   Div-Curl Boundary Value Problems.

        In collaboration with James C. Alexander, we wrote a number of papers on the use of variational methods to obtain well-posedness results for div-curl systems for various types of boundary conditions. This is a degenerate elliptic system and the uniqueness, or otherwise, of solutions depends on the toplogogy of the domains and the boundary data. See

  1.   (with J.C. Alexander), "L2-well-posedness of Planar div-curl Problems", Archive Rational Mechanics and Analysis 160, (2001), 91-134.

      2.   (with J.C. Alexander), "L2-well-posedness of 3d-div-curl Boundary value Problems",  Quarterly of  Applied Mathematics, V63, (2005), 479-508.

      3.   (with J.C. Alexander) "Finite Energy Solutions of Mixed 3D div-curl Systems", Quarterly of Applied Mathematics, 64, (2006), 335-357.

       Recent publications on these topics include

  1. "Electrostatic Approximation of Vector Fields", to appear in "System Modeling and Optimization", ed Bociu, Desideri and Habal, Springer.  [pdf]
  2.  "Bounds and Regularity of Solutions of Planar Div-Curl Problems", Quarterly of Applied Math, 75, (2017), 502-52.
  3.   (with Douglas Simpkins), "Spectral Representations and Approximations of Divergence-free Vector Fields", Quarterly of Applied Mathematics 74, (2016), 429-441.

  Some material related to the recent paper with D.R. Simpkins on the Representation and Approximation of Divergence-free Vector Fields is available here.


C.       Approximations, Eigenproblems and Inequalities.

         Many of the above results are related to the analysis of specific eigenproblems and the eigenfunctions provide bases that may be used for approximation and representation theorems.  

  1.   (with M. Cho), "Steklov  Approximations of Harmonic Boundary Value Problems", J. of Computational & Applied Matheatics, 321, (2017) 302-313.
  2. (with M. Rivas), "Laplacian Eigenproblems on Product Regions and Tensor Products of Sobolev Spaces ", J. Math Anal & Appns, 435 (2016), 842-859.
  3. (with M. Rivas), "Unconstrained Variational Principles for Linear Elliptic Eigenproblems", ESAIM, Control and Calculus of Variations, 21, (2015), 165-189.
  4. (with M. Cho), "Boundary Integrals and Approximations of Harmonic Functions", Numerical Functional Analysis & Appns, 36, (2015) 687-703.

      5.  "Parametric Dependence of Boundary Trace Inequalities," Proc Conf on Differential and Difference Equations, Springer Proceedings in Mathematics and Statistics, V 47, (2013), 249-254.
      6.    "Sharp Boundary Trace Inequalities", Proc Royal Society Edinburgh A, 144 (2014), 1-12.
      7.   "Bases and Comparison results for Linear Elliptic Eigenproblems",  J. Math Anal & Applications, 390, (2012),   394-406. A correction is here .



updated August 2017..

 

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