Scientific Publications: Giles Auchmuty.
The publications below are arranged by subject area. Some
papers are listed under more than one heading. Reprints of most of
the following papers are available from the author. Send e-mail to
auchmuty@uh.edu for copies or further information.
A. Steklov
Eigenproblems, Linear Boundary Value Problems and Harmonic
Functions.
1. "Steklov Eigenproblems and the Representation
of Solutions of Elliptic Boundary Value Problems", Numerical
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 Anal., 41, (2009) 1994-2009.
4. "Finite Energy Solutions of
Self-adjoint Elliptic Mixed Boundary Value
Problems", Math. Methods in the Applied Sciences,
33, (2010),
1446-1462.
5. (with P. Kloucek), ''Spectral Solution of
Self-adjoint Elliptic Problems with Immersed Interfaces",
Applied Mathematics and
Optimization. {\bf 64}, (2011), 311-338.
6. "Bases and
Comparison Results for Linear Elliptic Eigenproblems",
J. Math. Anal. Appn, 390 (2012), 394-406.
See corrections
are here.
7. (with Q. Han), "Spectral Representations of Solutions of
Linear Elliptic Equations on Exterior Regions",
J Math Anal
Appns 398 (2013), 1-10.
8. (with Q. Han),
"Representations of Solutions of Laplacian
Boundary Value Problems on Exterior Regions",
Applied
Mathematics and
Optimization, 69, (2014), 21-45.
9. (with M. A. Rivas),
"Unconstrained Variational Principles for Linear
Elliptic Eigenproblems", J. Math. Anal. &
Appns,
ESIAM, Control and Calculus of
Variations 21, (2015), 165-189.
10. (with Q. Han),
"Well-posedness and Approximation of Solutions of
Linear Divergence-form Elliptic Problems on
Exterior
Regions",
Mathematical Methods in the Applied Sciences, 38,
(2015), 1867-1875.
11. (with M. Cho) "Boundary Integrals
and Approximations of Harmonic Functions",
Numerical Functional Analysis
and
Applications, 36, (2015), 687-703.
12. "The SVD of the Poisson
kernel," J. Fourier Analysis and Applications, 23,
(2017) 1517-1536.
13. "Steklov Representations of Green's Functions for
Laplacian Boundary Value Problems", Applied Mathematics and
Optimization, 74, (2016) 1-23.
14. (with M. Cho) "Steklov Approximations of Harmonic
functions on Planar Regions", J. Comp. & Applied Math,
321, (2017), 302-313.
B. Bifurcation Theory,
Bifurcating waves and Reaction-Diffusion Equations.
- "Lyapunov Methods and Equations of
Parabolic Type", Springer Verlag Lecture Notes in
Mathematics, Vol. 332, (1973), 1-14.
- (with G. Nicolis), "Dissipative Structures,
Catastrophes and Pattern Formation: A Bifurcation Analysis",
Proc. National Academy of Science, 71, (1974), 2748-2751.
- (with G. Nicolis), "Bifurcation Analysis
of Reaction-Diffusion Equations, I." Bull. Mathematical
Biology, 37, (1975), 323-365.
- (with G. Nicolis) "Bifurcation Analysis of
Reaction-Diffusion Equations, III. Concentration
Oscillations", Bull. Mathematical Biology, 38, (1975),
325-350.
- "Positivity for Parabolic and Elliptic
Systems", Proc. Royal Society of Edinburgh, 79A, (1977),
183-191.
- "Bifurcating Waves", Annals New York
Academy of Science, (Proceedings Conference on Bifurcation
Theory & its Applications), 316, (1979), 263-278.
- "Qualitative Effects of Diffusion in
Chemical Systems", Lectures on Mathematics in the Life
Sciences, Vol. 10, American Math. Soc. (1978), 49-99.
- (with J.C. Alexander), "Global Bifurcation
of Waves", Manuscripta Mathematica, 27, (1979), 159-166.
- "Bifurcation Analysis of
Reaction-Diffusion Equations, IV. Size Dependence", in
Instabilites, Bifurcations & Fluctuations in Chemical
Systems, (1982), 3-31, U. Texas Press.
- "Bifurcation Analysis of
Reaction-Diffusion Equations, V. Rotating Waves on A Disc"
in Partial Differential Equations and Dynamical Systems, ed.
W.E. Fitzgibbon, Research Notes in Mathematics, Vol. 101,
(1984), 35-63, Pitman.
- (with J.C. Alexander) "Global Bifurcation
of Phase-Locked Oscillators", Archive Rational Mechanics
& Analysis, 93, (1986) 253-270.
- (with J.C. Alexander) "Bifurcation
Analysis of Reaction-Diffusion Equations, VI. Multiply
Periodic Traveling Waves", SIAM J. Math. Anal. 19, (1988)
100-109.
- (with J.G. Alford) "Rotating Wave Solutions
of FitzHugh Nagumo Equations", J of Mathematical
Biology, 53, (2006) 797-820.
C.
Convex Analysis, Minimax Theory, Nonlinear PDE and
Variational Methods.
- "Duality for Non-Convex Variational
Principles", J. Differential Equations, 50, (1982), 80-145.
- "Variational Principles for Periodic
Solutions of Autonomous Ordinary Differential Equations", in
Nonlinear Oscillations in Chemistry and Biology, Lecture
Notes in Biomathematics, Vol. 66, (1986) Springer Verlag pp
252-260.
- "Variational Principles for Operator
Equations and Initial Value Problems", J. Nonlinear
Analysis, 12, (1988) 531-564.
- (with E. Dean, R. Glowinski and S.C.
Zhang) "Optimal Control Methods for finding Periodic
Solutions of Differential Equations" in "Control Problems
for Systems Described by Partial Differential Equations and
Applications," ed. I. Lasiecka and R. Triggiani, Lecture
Notes in Control and Information Science, Vol. 97, (1987),
pp 64-89, Springer Verlag.
- "Variational Principles for Variational
Inequalities", Nonlinear Funct. Anal. and Optimization, 10,
(1989) 863-874.
- "Minimax Theorems and Differential
Inclusions", Proc. Int. Conference on Differential
Equations, Edinburg, Tx., Research Notes in Math Vol 273,
(1992), Pitman 6-10.
- "Saddle-points and Existence-Uniqueness
for Evolution Equations", Differential & Integral
Equations, 6, (1993), 1161-1171.
- "Min-max Methods for Non-Potential
Operator Equations", in Optimization Methods in Partial
Differential Equations, ed Cox & Lasiecka,
Contemporary Mathematics, Vol 209, American Mathematical
Society (1997), 19-28.
- "Control-theoretic Methods for Computing
Special Solutions of Differential Equations", in Computational
Science in the 21st Century, ed Periaux et al. John
Wiley (1997), 655-659.
- "Variational Principles and Residual Bounds
for Non-Potential Equations", Advances in Applied
Mathematics and Optimization, ed Gao and Sherali, Springer
(2009), 13-24.
- "Variational Principles for Initial Value
Problems", Contemporary Mathematics V426, AMS (2007) 45-56.
- (with Q. Han), "p-Laplacian Boundary Value
Problems on Exterior Regions", J. Math Analysis &
Applications, 417,
(2014) 260-271.
- "Variational Principles for
Advection-Diffusion Equations", Computers and Mathematics
with Applications, (to appear, available online
23.10.17).
D. Div-Curl Boundary Value
Problems, Fluid Mechanics and Vector Analysis.
- "Bounding Flows in
Turbulent Convection", Archive for Rational Mechanics &
Analysis, 51, (1973), 228-238.
- "Orthogonal Decompositions and Bases for
3-d Vector Fields", Numerical Functional Analysis and
Optimization, 15, (1994), 455-488.
- "Potential Representations of
Incompressible Vector Fields", in Nonlinear Problems in
Applied Mathematics, ed Angell et al, SIAM, (1995),
43-49.
- "Reconstruction of the Velocity from the
Vorticity in 3-d Fluid Flows", Proc. Royal Society London,
A454, (1998), 607-630.
- (with J.C. Alexander),
"L2-well-posedness of Planar div-curl Problems", Archive
Rational Mechanics and Analysis 160, (2001), 91-134.
- "The Main Inequality of Vector Field
Theory", Math. Modeling and Methods in Applied
Sciences, 14, (2004) 79-103. [pdf]
- (with J.C. Alexander), "L2-well-posedness of
3d-div-curl Boundary value Problems", Quarterly of
Applied Mathematics, V63, (2005), 479-508. [pdf]
- "Divergence L2-coercivity Inequalities",
Numerical Functional Analysis and Applications, 27, (2006),
499-516. [pdf]
- "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.
- (with J.C. Alexander), "Finite Energy
Solutions of Mixed 3D div-curl Systems", Quarterly of Applied
Mathematics, 64, (2006), 335-357. [pdf]
- (with P. Kloucek), "Generalized Harmonic
Functions and the Dewetting of Thin Films", Applied Functional
Analysis and Optimization, 55, (2007), 145-161.
- " Boundary Control of an electrostatic Field
", in "System Modeling and Optimization", ed Bociu, Desideri
and Habbal, Springer (2017) 89-94.
- (with D.S. Simpkins), Spectral Representations
and Approximations of Divergence-free Vector Fields",
Quarterly of Applied Math, (74) (2016) 429-441.
- "Bounds and Regularity of
Solutions of Planar Div-curl Problems", Quarterly
of Applied Math., 75,
(2017), 505-524.
E.
Eigenvalue Problems and Linear Elliptic Boundary Value Problems.
- "Dual Variational Principles for Eigenvalue
Problems" in Nonlinear Functional Analysis and its
Applications (ed. F.E. Browder), Proc. Symposia in Pure
Mathematics, Vol. 45, Part 1, pp 55-72, AMS 1986.
- "Unconstrained Variational Principles for
Eigenvalues of Real Symmetric Matrices," SIAM J. Applied Math,
20, (1989) 1186-1207.
- "Variational Principles for Eigenvalues of
Nonsymmetric Matrices," SIAM J. Matrix Anal. & Applns, 10,
(1989) 105-117.
- "Variational Principles for Eigenvalues of
Compact Operators," SIAM J. Applied Math., 20, (1989),
1321-1335.
- "Globally, and Rapidly, Convergent
Algorithms for Symmetric Eigenvalue Problems", SIAM J. Matrix
Anal. and Applications, 12, (1991), 690-706.
- (with W.Y. Jia), "Iterative Methods for
Hartree Eigenproblems", Mathematical Modeling and Numerical
Analysis, 28, (1994), 576-610.
- "Variational Principles for Self-Adjoint
Elliptic Eigenproblems", in Nonsmooth / Nonconvex Mechanics,
ed Gao, Ogden & Stavroulakis, pp 15-42, Kluwer 2001.[pdf]
- "Algorithms and Merit Functions for the
Principal Eigenvalue", in Advances in Convex Analysis
and Global Optimization, ed Hadjisavvas & Pardolas,
Kluwer, (2001), 235-244.
- "Bases \& Comparison Results for
Linear Elliptic Eigenproblems", J. Math Anal &
Appns., 390, (2012), 394-406.
- (with Mauricio A. Rivas) "Unconstrained
Variational Principles for Linear Elliptic Eigenproblems"
ESIAM, Control and Calculus of Variations, 21, (2015),
165-189.
- (with M.A. Rivas), "Laplacian Eigenproblems on
Product Regions and Tensor Products of Sobolev
Spaces", J. Math Anal & Appns, 435,
(2016) 842-859.
F. Algorithms
and Numerical Simulation.
- "The Numerical Solution of the Equations for
Rotating Stars", Springer Verlag Lecture Notes in Mathematics,
Vol. 430, (1974) 28-60.
- "Numerical Models of Rotating Stars",
Springer Verlag Lecture Notes in Physics, Vol. 35, Proceedings
of the Fourth International Conference on Numerical Methods in
Fluid Dynamics (1975) pp. 40-44.
- "Numerical Calculation of Rotating Stars",
Méthodes Numériques en Mathématiques Appliquées, Les Presses
de L'Université de Montréal, (1977), 9-26.
- "Duality Methods for Non-convex
Optimization", Numerical Functional Analysis and Optimization,
10, (1989), 211-264.
- "Duality Algorithms for Smooth,
Unconstrained Optimization", in Computational Solution of
Nonlinear Equations, Proc. Symposia in Applied Mathematics,
26, (1990) AMS, 15-29.
- "Globally, and Rapidly, Convergent
Algorithms for Symmetric Eigenvalue Problems", SIAM J. Matrix
Anal. and Applications, 12, (1991), 690-706.
- "A Posteriori Error Estimates for Linear
Equations," Numerische Mathematik, 61, (1992), 1-6.
- "Lagrangean Variational Principles and
Algorithms for the Equilibria of Rotating Self-Gravitating
Fluids", Proc. of Conference on Free and Moving Boundary
Problems, CMA, Canberra, 30, (1992), 18-41.
- "Saddle Point Methods and Algorithms for
Solving Non-symmetric Linear Equations", Numer. Functional
Anal. & Optimiz. 16, (1995), 1127-1142.
- (with C. Zhai), "Computation of Rotating
Wave Solutions of Reaction-Diffusion equations on a Circle",
East-West J. of Numerical Analysis, 4, (1996), 151-163.
G. Gravitation
and Models of Rotating Self-gravitating Fluids.
- (with R. Beals), "Models of Rotating
Stars", Astrophysical J. May 1, 1971, L79-82.
- (with R. Beals), "Variational Solution of
some Nonlinear Free Boundary Problems", Archive for Rational
Mechanics & Analysis, 43 (1971), 255-271.
- "Existence of Axisymmetric Equilibrium
Figures", Archive for Rational Mechanics & Analysis, 65,
(1977), 249-261.
- "Models of Rotating Self-gravitating
Liquids", Trends in Applications of Pure Mathematics to
Mechanics, Vol.III, (1981), 22-39, Pitman.
- "The Global Branching of Rotating Stars",
Archive for Rational Mechanics and Analysis, 114, (1991)
179-194.
- "An Inequality in Gravitation Theory",
SIAM Review, 39, (1997) 127-135.
H.
Mathematical Biology, Chemistry and Finance.
- (with G. Nicolis), "Dissipative Structures,
Catastrophes and Pattern Formation: A Bifurcation Analysis",
Proc. National Academy of Science, 71, (1974), 2748-2751.
- "Qualitative Effects of Diffusion in
Chemical Systems", Lectures on Mathematics in the Life
Sciences, Vol. 10, American Math. Soc. (1978), 49-99.
- (with W.Y. Lee and R. Rao), "Option Pricing
in a lognormal Securities Market with discrete trading", J.
Financial Economics, 9, (1981), 75-101.
- (with J. Chadam, E. Merino, P. Ortoleva and
E. Ripley) "The Structure and Stability of Propagating Redox
Fronts", SIAM J. of Applied Math, 46, (1986), pp 588-604.
- (with P. Ortoleva, J. Chadam, J. Hettmer, E.
Merino, C. Moore and E. Ripley), "Redox Front Propagation and
Banding Modalities", Physica D. 19D (1986), pp 334-358.
- (with J. Chadam, E. Merino, P. Ortoleva and
E. Ripley) "Moving Redox Fronts and Worm-hole Cave Cascades"
Proc. DOE Argonne Conf. on Nonlinear Problems in Energy
Research, (1984).
- (with 9 coauthors) "Kinetic Modeling of
Flow-Reaction Instabilities in Porous Rocks: Application to
Uranium Roll Fronts", Proceedings 4th Int. Symp. on Water/rock
Interaction, Misasa, Japan, (1984).
- (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
- (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.
I. Inequalities.
- (with D. Bao), "Harnack-type Inequalities
for Evolution Equations", Proc. American Math. Soc., 122,
(1994) 117-130.
- (with J. He), "An Integral Inequality in
Population Modeling", Solution of problem 95-13, SIAM Review
40, (1998), 710-713.
- (with J. He), "An Inequality for Periodic
Functions and a Quadratic Imbedding Result", Numerical
Functional Analysis and Applications, 21, (2000), 781-789.
- "The Main Inequality of Vector Field Theory",
Math. Modeling and Methods in Applied Sciences, 14,
(2004) 79-103.
- "Optimal Coercivity Inequalities in W^(1,p)",
Proc. Royal Society of Edinburgh, 135A, (2005), 915-933.
- (with B. Emamizadeh and M. Zivari),
"Dependence of Friedrichs' Constant on Boundary
Integrals", Proc. Royal Society Edinburgh, 135A, (2005),
935-939.
- "Divergence L2-coercivity Inequalities",
Numerical Functional Analysis and Applications, 27, (2006),
499-516.
- "Imbeddings and Bounds for
Functions with L^q Laplacians", J. Math Anal, Appns, 383,
(2011), 25-34. see also the comment .
- "Sharp Boundary Trace Inequalities", Proc
Royal Soc. Edinburgh, A 144 (2014) 1-12.
- "Parametric Dependence in Boundary Trace
Inequalities", in Proceedings Conference on Differential and
Difference Equations, ed Pinellas et al, Springer (2013)
50-55.
updated February 25, 2018.