Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T04:34:00.626Z Has data issue: false hasContentIssue false
  • English
  • Français

INTERPOLATION UNDER A GRADIENT BOUND

Published online by Cambridge University Press:  01 August 2009

GUNNAR ARONSSON
GUNNAR ARONSSON*
Affiliation:
Department of Mathematics, Linköping University, S-58183 Linköping, Sweden (email: guaro@mai.liu.se)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with the interpolation of given real boundary values into a bounded domain in Euclidean n-space, under a prescribed gradient bound. It is well known that there exist an upper solution (an inf-convolution) and a lower solution (a sup-convolution) to this problem, provided that a certain compatibility condition is satisfied. If the upper and lower solutions coincide somewhere in the domain, then several interesting consequences follow. They are considered here. Basically, the upper and lower solutions must be regular wherever they coincide.

Keywords

eikonalinfimal convolutionsupremal convolutioninterpolationuniqueness setweighted lengthshortest curveoptimal controlmaximum principlesemiconcave
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 19 - 35
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Akhiezer, N. I., The Calculus of Variations (Blaisdell, New York, 1962).Google Scholar
[2]Aronsson, G., ‘Extension of functions satisfying Lipschitz conditions’, Ark. Mat. 6(28) (1967), 551–561.Google Scholar
[3]Aronsson, G., ‘On the partial differential equation u x2u xx+2u xu yu xy+u y2u yy=0’, Ark. Mat. 7(28) (1968), 395–425.Google Scholar
[4]Aronsson, G., Crandall, M. and Juutinen, P., ‘A tour of the theory of absolutely minimizing functions’, Bull. Amer. Math. Soc. 41(4) (2004), 439–505.Google Scholar
[5]Aubert, G. and Kornprobst, P., Mathematical Problems in Image Processing (Springer, Berlin, 2002).Google Scholar
[6]Bryson, A. E. Jr and Ho, Y.-C., Applied Optimal Control (Ginn and Company, Waltham, MA, 1969).Google Scholar
[7]Buttazzo, G., Giaquinta, M. and Hildebrandt, S., One-dimensional Variational Problems (Oxford University Press, Oxford, 1998).Google Scholar
[8]Cesari, L., ‘Existence theorems for optimal solutions in Pontryagin and Lagrange problems’, SIAM J. Control, Ser. A 3(3) (1966), 475–498.Google Scholar
[9]Crandall, M., ‘An efficient derivation of the Aronsson equation’, Arch. Ration. Mech. Anal. 167(44) (2003), 271–279.Google Scholar
[10]Champion, T. and DePascale, L., ‘Principles of comparison with distance functions for absolute minimizers’, J. Convex Anal. 14(3) (2007), 515–541.Google Scholar
[11]Clarke, F. H., Ledyaev, Y. S., Stern, R. J. and Wolenski, P. R., Nonsmooth Analysis and Control Theory (Springer, Berlin, 1998).Google Scholar
[12]Cannarsa, P. and Sinestrari, C., Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control (Birkhäuser, Basel, 2004).Google Scholar
[13]Durou, J. D., Falcone, M. and Sagona, M., ‘A survey of numerical methods for shape from shading’, IRIT Rapport 2004-2-R, Université Paul Sabatier, Toulouse, 2004.Google Scholar
[14]Jensen, R., ‘Uniqueness of Lipschitz extensions. Minimizing the sup norm of the gradient’, Arch. Ration. Mech. Anal. 123(1) (1993), 51–74.Google Scholar
[15]Lions, P. L., Generalized Solutions of Hamilton–Jacobi Equations, Research Notes in Mathematics, 69 (Pitman Publishing, Boston, 1982).Google Scholar
[16]Lee, E. B. and Markus, L., Foundations of Optimal Control Theory (Wiley & Sons, New York, 1967).Google Scholar
[17]Pinch, E. R., Optimal Control and the Calculus of Variations (Oxford University Press, Oxford, 1993).Google Scholar
[18]Pontryagin, L. S., Boltyanski, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., The Mathematical Theory of Optimal Processes (Interscience Publishers, New York, 1962).Google Scholar
[19]Sethian, J. A., Level Set Methods (Cambridge University Press, Cambridge, 1996).Google Scholar
[20]Yu, Y., ‘Variational problems and Aronsson equations’, Arch. Ration. Mech. Anal. 182 (2006), 153–180.CrossRefGoogle Scholar