Solvability of elliptic partial differential equations with rough coefficients; the boundary value problems

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics

Abstract

The proposed research aims to study elliptic partial differential equations. Partial differential equations are used to mathematically describe behaviour of many real life phenomena and arise practically everywhere, such as in physics, material science, geometry, probability and many other disciplines.

In many real life models such as in material science or physics the coefficients can be discontinuous (modelling impurities in the material or cracks) and so it makes sense to study equations with
low regularity coefficients. Here the discontinuity of coefficients is the mathematical expression of the fact that many materials contain impurities (foreign objects) that somewhat change the properties of studied objects. It is therefore very important to consider these situations mathematically and understand way of solving such equations. This is what we are going to do in our research.

As the name of this project suggests we are going to focus on three basic types of boundary value problems. Our knowledge about different boundary value problems is quite uneven, in particular one type of boundary value problem is much better explored than the other two. We aim to remedy this situation and bring our knowledge about these boundary value problems to approximately same level.

Planned Impact

The proposed research falls within the scope of basic research in Mathematics hence it would be unrealistic to project any immediate economic and social impacts. For these reasons most of the impact
our research will have is academic. The research however does have good potential in a medium term to find applications outside mathematics either in physics or material sciences. For this reason we plan to engage in communications about our results with people not only in mathematics but also in other fields. We will employ several ways of doing that. We plan to make our results avaliable early on personal webpages, Arxiv and in later stage by publishing them in peer-reviewed journals. We will attend research conferences presenting the results in talks and personal communications. Finally, thanks to ICMS (the International Centre for Mathematical Sciences) Edinburgh has a wide and constant stream of research visitors. We plan to engage them, let them know about our results and potential implication for their own work. Many of these researchers are from areas outside mathematics but working on problems where analysis and PDEs play important role.

The ICMS also provides with opportunities to organize workshops/research conferences in Edinburgh. We plan to organize one such event during the duration of our grant on topic related to this research. Such event should significantly increase visibility of this research topic increasing the likelihood of impacts.

Publications

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Dindos, M. (2017) BMO solvability and the $A_\infty$ condition for second order parabolic operators in Annales de l'Institut Henri Poincaré, to appear

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Dindoš M (2017) Boundary Value Problems for Second-Order Elliptic Operators Satisfying a Carleson Condition in Communications on Pure and Applied Mathematics

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Dindos, M. The Boundary value problems for second order elliptic operators satisfying a Carleson condition in Communications on Pure and Applied Mathematics, accepted. Posted on arXiv:1301.0426

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Hwang, S. (2015) Hölder continuity of a bounded weak solution of generalized parabolic p-Laplacian equations in Electronic Journal of Differential equations

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Dindos M. (2017) The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition in Revista Math. Iberoamericana to appear, posted on arXiv:1402.0036

 
Description In recent years substantial progress has been achieved in our understanding of scalar elliptic PDEs with rough coefficients and associated boundary value problems. The results of this grant have significantly contributed to this understanding, where we have successfully resolved several open questions for certain important boundary value problems.

The motivation to study equations with rough (or low regularity) coefficients is as follows. In many "real life" models, such as in materials science, the coefficients can be discontinuous (for example modeling impurities in the materials or cracks).

Another topics that was investigated were PDEs on domains with rough boundaries. PDEs on such domains occur often; for example polyhedral boundaries (whose simplest representative in two dimensions is a rectangle) occur in physics or engineering.
Exploitation Route As stated above, there are potential applications of our results in engineering (models of materials with impurities and cracks), physics (linear and nonlinear elasticity). Some further future applications are in extending these results to non-linear theory. For example in efforts to tackle nonlinear (such as quasi-linear elliptic) equations we often encounter coefficients of the highest order terms in the equation(s) that depend themselves on the solution or its derivatives. Hence a priori very little is know about the regularity of such coefficients. It is therefore extremely useful to have available linear theory for these PDEs with coefficients that have minimal smoothness.

In the summer 2015 I have organised a 1 week conference at the ICMS in Edinburgh that was purposefully dedicated to the topics of this proposal. The aim of the conference was to bring together researchers working in the area with aim to exchange ideas and to dissipate the results of their research as well as mine. This way the results of the research undertaken by me and my collaborators under auspices of this grant are not only published (which usually takes some time) but are directly and quickly communicated to relevant people who can take the findings forward in the future.
Sectors Aerospace, Defence and Marine,Education,Other

URL http://arxiv.org/find/math/1/au:+Dindos_M/0/1/0/all/0/1