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

The problem is given the divergence and the curl of a vector field in a given region, what extra data is necessary in order to find the field? When the region is  two-dimensional, this is a system of two linear equations in two unknowns which has been exhaustively studied.  However , the world is three dimensional so real physical problems  lead to four equations for 3 the unknown components of the field and this problem is always an over-determined linear system of first order linear PDEs. The system is solvable only when certain compatibility conditions hold. These compatibility conditions depend not only on the equations but also on terms arising from the boundary conditions and are derived and described in 4 and 7 below for the case of a bounded region with a reasonable boundary.

Paper 1 below describes the solvability of these problems when the velocity of the fluid is zero (or given) at eavery point of the boundary. That is there are 3 boundary conditions at each point of the boundary.

In electromagnetic field theory, however, the boundary conditions typically may be
(i)   normal component of the field is prescribed at each point on the boundary (1 boundary condition), or
(ii)  tangential component of the field is prescribed at each point of the boundary (2 b.c.s), or even
(iii)  at some points the normal component is prescribed and at others the tangential component is  given.
Each of these problems has different compatibility conditions, different existence-uniqueness criteria and
and the associated "well-posed" problem  may depend on the differential topology of the region (the number 
of holes and handles in the region). These results (for finite energy solutions) are described in papers 4 and 7.

As with all elliptic problems, a basic issue is the proof of coercivity for certain quantities. Results of this type are proved in 3 and 5.

  1. "Reconstruction of the Velocity from the Vorticity in 3-d Fluid Flows", Proc. Royal Society London, A454, (1998), 607-630.
  2. (with J.C. Alexander), "L2-well-posedness of Planar div-curl Problems", Archive Rational Mechanics and Analysis 160, (2001), 91-134.
  3. "The Main Inequality of Vector Field Theory", Math. Modeling and Methods in Applied Sciences,  14, (2004) 79-103.  [pdf]
  4. (with J.C. Alexander), "L2-well-posedness of 3d-div-curl Boundary value Problems",  Quarterly of  Applied Mathematics, V63, (2005), 479-508. 
  5. "Divergence L2-coercivity Inequalities", Numerical Functional Analysis and Applications, 27, (2006), 499-516.
  6. "The Stokes Basis for 3d Incompressible Fluid Flows". in Free and Moving Boundaries, Analysis, Simulation and Control, ed R. Glowinski and J-P Zolesio, Lecture Notes in Pure and Applied Mathematics, V 252, Chapman and Hall/CRC (2007), 215-222.  
  7. (with J.C. Alexander), "Finite Energy Solutions of Mixed 3D div-curl Systems", Quarterly of Applied Mathematics, 64, (2006), 335-357.

 

B.     Linear Elliptic Boundary Value Problems and Trace Results.

All the papers below treat problems where boundary terms are of interest. Papers 2 and 3 describe W^(1,p) coercivity results that include boundary integral terms (as does 5 in the previous section). The other papers listed here all describe results that involve Steklov eigenproblems and their uses.

A Steklov eigenproblem  is the problem of finding those lambda such that there is a non-zero solution of
Lu = 0 in a region Omega subject to        D_nu u = lambda rho u on the boundary.
That is the eigenparameter is in the boundary data - and the equation is homogeneous. In 1, there are a variety of results about these problems for second order elliptic operators including some completeness theorems. These results lead to representation theorems for the solutions of Lu = 0 in Omega subject to different boundary conditions. In 4, it is shown that the solutions of the harmonic Steklov problem for a region provides a natural description of the Hilbert trace spaces on the boundary. This enables the description of trace spaces under weak regularity of the boundary (weaker than Lipschitz domains), including explicit formulae for the H^{1/2} and H^{-1/2} inner products. In 5 these results are used to describe a spectral approach to solving Laplace's equation with discontinuous boundary data on a 2d region.

Paper 6 proves that certain Hilbert spaces of real harmonic functions are  reproducing kernel Hilbert spaces, provides expressions for the reproducing kernel  and describes  other properties of these spaces. This extends similar results of J-L. Lions, J. Peetre and others.

Papers 7 and 8 in the listing describe spectral representations of solutions of various non-standard boundary value problems for 2nd-order self-adjoint linear elliptic equations on bounded regions. In each case special eigenproblems are introduced to devlop these descriptions.

  1. "Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems", Num. Functional Analysis and Optimization, 25, (2004), 321-348.  
  2. "Optimal Coercivity Inequalities in W^(1,p)", Proc. Royal Society of Edinburgh, 135A, (2005), 915-933.  
  3. (with B. Emamizadeh and M. Zivari), "Dependence of Friedrichs' Constant on Boundary Integrals",  Proc. Royal Society Edinburgh, 135A, (2005), 935-939. 
  4. "Spectral Characterization of the Trace Spaces H^s(bdy)", SIAM J of Math. Anal., 38, (2006) 894-905. 
  5. (with P. Kloucek), "Generalized Harmonic Functions and the Dewetting of Thin Films", Applied Functional Analysis and Optimization, 55, (2007), 145-161.
  6. "Reproducing Kernels and Hilbert Spaces of Real Harmonic Functions", SIAM J Math Analysis, 41, (2009), 1994-2009.
  7. "Finite Energy Solutions of Mixed Elliptic Boundary Value Problems", Math Methods for the Applied Sciences, 33, (2010) 1446-1462. [pdf]
  8. (with P. Kloucek), "Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces", Applied Math and Optimization 64, 2011, 311-338.
  9. "Imbeddings and Bounds for Functions with L^q-Laplacians", J. Math Anal & Applications, 383, (2011), 25-34.. 
  10. "Parametric Dependence of Boundary Trace Inequalities," Proc Conf on Differential and Difference Equations, (Springer), to appear. 
  11. "Bases and Comparison results for Linear Elliptic Eigenproblems", J. Math Anal & Applications, J. Math Anal & Applications, 390, (2012), 394-406.
  12. "Sharp Boundary Trace Inequalities", submitted. (11 pages).

 

C.     Variational Methods for solving equations.

These papers describe variational principles for finding solutions of some classes of "non-potential equations". The first is for certain classes of finite dimensional equations. The second is for some initial value problems for ordinary differential equations.

  1. "Variational Principles and Residual Bounds for Non-Potential Equations", in Advances in Applied Mathematics and Global Optimization, ed Gao and Sherali, Springer Advances in Mechanics and Mathematics Vol 17, (2009) 13 - 23.  
  2. "Variational Principles for Initial Value Problems", Contemporary Mathematics V426, AMS (2007) 45-56. A correction is available here [pdf]

 

D.    Mathematical Biology

The following papers are with former graduate students (and others) and involve models where bifurcation theory was used to study specific phenomena.

  1. (with M.N. Obeyesekere, E.S. Tecarro and S.O. Zimmerman), "A model of cell cycle behavior dominated by kinetics of a pathway stimulated by growth factors", Bull. Math. Biology, 61, (1999), 917-934
  2. (with M.N. Obeyesekere and E.S. Tecarro), "Analysis of a model of the Mammalian Cell cycle's G1 phase", Nonlinear Analysis and Applications, Real World Applications, 4, (2003), 87-107.
  3. (with J.G. Alford) "Rotating Wave Solutions of FitzHugh Nagumo Equations", J of Mathematical  Biology, 53, (2006) 797-820.



updated May 2011..

 

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