The European Journal of Applied Mathematics and Cambridge University Press are pleased to award the 2016 John Ockendon Prize to S. J Chapman and S. E McBurnie for their winning article ‘Integral constraints in multiple-scales problems’ published in EJAM’s Special Anniversary Issue, October 2015.

The John Ockendon Prize, named after the founding editor of EJAM was launched this year and will continue to run every two years. The winning paper was selected after Editorial Board Members of EJAM nominated and voted on papers published in 2014 and 2015 that demonstrated excellent mathematical research relevant to real applications. The short-listed articles are available to read in full here until 31st August 2016.

On receiving his award at the British Applied Mathematical Colloquium, Professor S. J Chapman shared his thoughts on winning: “John Ockendon was my DPhil supervisor and has been a valued colleague and mentor for many years. The paper was published in a special issue of EJAM to mark his 75th birthday. I was therefore particularly delighted to learn that it had been awarded the inaugural John Ockendon prize.”

Editorial board members Koby Rubinsten and Thomas P. Witelski provide detailed summaries of why they nominated this winning paper:
“Many scientific, medical and industrial problems involve materials whose properties vary on a very fine scale. For example we mention conductivity and elasticity in composite materials, such as those used in the aerospace industry, signal propagation in the heart muscle and other tissues, filtration of water or oil in rocks. The list is essentially endless. Solving exactly the equations that describe such materials is a formidable task, even with the use of very powerful computers. Moreover, practically, what matters is the overall behaviour of these materials and not their response at the very fine scale.

“To handle such problems mathematicians developed over the last half century a powerful method, called homogenization, where the equations are averaged over the local, fine, scale. Therefore it is required to solve much simpler equations on a global, macroscopic scale, where the microscopic geometry is represented by effective parameters. This technique has met with tremendous success, enabling the solution of many problems, including better oil and gas exploration methods, the design of sophisticated optical devices, improved understanding of tumour growth, propagation of dental caries and much more.”

Rubinsten continues “Still some fundamental issues in the theory of homogenization were unresolved until recently. In particular, the theory contained a strange paradox when handling problems that have widely varying geometric scales, and at the same time widely varying material properties. In their award winning paper ‘Integral constraints in multiple-scales problems’, S. J Chapman and S.E Mcburnie resolved this paradox. Carrying out a careful and delicate analysis of the homogenization method for such cases, they showed how to handle simultaneously these two limits. The analysis of Chapman and Mcburnie has thus become a basic component of the half century old highly successful method of homogenization.”

Thomas P. Witelski further adds “The article by Chapman and McBurnie makes a valuable contribution to the analysis of problems for complex materials. Building on classical homogenization theory from multiple scale asymptotics, the authors consider a set of problems modelling physical systems with high contrast differences in materials properties. Motivated by a longstanding claim on how one such problem on acoustic waves in bubbly fluids should be solved, the authors have made clear how the various limits in the system can be handled. In addition to advancing understanding of problems with strong microstructure, the paper provides an excellent introduction to homogenization theory, building up from a simple illustrative one-dimensional example to the full problem in three dimensions.”

The next John Ockendon prize will be awarded in 2018 covering articles published in 2016 and 2017. We are delighted to provide complimentary access to all papers nominated for this prize, in addition to the winning paper.