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# A functional analytic approach for the analysis of nonlinear transmission problems

## Periodic Reporting for period 1 - FAANon (A functional analytic approach for the analysis of nonlinear transmission problems)

Reporting period: 2015-12-01 to 2017-11-30

"We have addressed nonlinear contact problems in domains with singular boundary perturbations. To do so we have implemented a new method called Functional Analytical Approach (FAA in short). A special attention has been paid to the case of perforated and periodically perforated domains with shrinking holes and inclusions.

In physics and material engineering elliptic boundary value problems with contact conditions defined on 2-D or 3-D domains play an important role. For example, they appear in the models for elasto-plastic composites, for composite of materials with different thermal conductivities, and in permeable materials with porous inclusions. The classical theory of elliptic equations explains very well what we have to expect when the domains have a smooth boundary; however, natural domains are often non-smooth and may be the result of ""singular perturbations"" of more regular domains. The direct application of mathematical software based on numerical methods to such singularly perturbed problems often leads to poor accuracy and numerical instability. To overcome this difficulty, we must first perform a theoretical analysis to understand the effect of the perturbation and provide an effective computation strategy. In literature, the main approach adopted for this purpose is that of Asymptotic Analysis (AA). However, only few papers in AA have been dedicated to nonlinear problems, which are instead very natural in physics models. For this reason we have adopted a different approach: the FAA. Such approach has proven to be convenient in the treatment of nonlinear conditions and, in addition, permits to represent the effect of perturbations in terms of analytic functions.

The final goal within the action is the realization of a complete theory based on the FAA which will represent in future a fundamental tool for the analysis of perturbed boundary value problems, both at a theoretical level and in view of applications. To this end, the action has focused on 6 main objectives: i) to analyse existence and uniqueness of the solution for problems with nonlinear transmission conditions; ii) to develop mathematical instruments for the analysis of shape perturbations of possible inclusions; iii) to study non-ideal and nonlinear transmission problems in singularly perturbed domains; iv) to analyze the effective conductivity of dilute composites with imperfect contact at the interphase boundary important for applications; v) to set the basis for the implementation of numerical methods for the class of the problems analysed in this project; vi) to address shape perturbations in imperfect nonlinear contact problems of elasticity theory.

In conclusion, the action has proven the efficiency of the FAA in the study of singularly perturbed boundary value problems with linear and nonlinear boundary conditions. We have shown how to address existence and uniqueness issues that are inherent in such problems, also by applying results for small inclusions and perturbations. We have obtained mathematical results in potential theory that can be exploited in the study of shape perturbations. We have seen how to pass from the analytical results to the explicit computations in specific applications to effective properties of composites. Besides achieving most of the results which were planned, the action has set the basis for further research directions that the fellow will pursue together with mathematicians at Host (Aberystwyth University (AU), Wales, UK) and other European universities."
We refer to the objectives i)-vi) listed above:
i) the work on i) stems from a joint paper of the researcher of the fellow and G. Mishuris. We planned to investigate three different types of conditions. For two of them have obtained results that have been presented in joint papers with R. Molinarolo (PhD candidate at AU). One of the three conditions will require the introduction special techniques together with the FAA;
ii) the fellow has written a paper with M. Lanza de Cristoforis (University of Padova) and P. Musolino (AU) that has been submitted for the publication. The paper shows specific properties of volume potentials;
iii) the fellow has written three papers on this topic. A paper with P. Musolino has been published in a major mathematical journal (J. Differential Equations) and deals with the case of two inclusions that collapse to a point while shrinking one to the other at a slower rate. Another paper with L. Provenzano (EPFL, Lausanne, Switzerland) concerns the case when the inclusion, instead of collapsing to a point, fills the whole domain and has been submitted for the publication. Finally, a paper with V. Bonnaillie-Noël (CNRS, Ecole Supérieure, Paris), M. Dambrine (Université de Pau et des Pays de l'Adour, France), and P. Musolino concerns problem in a domain with a small shrinking inclusion that approaches the exterior boundary;
iv) the fellow is writing a paper together with P. Musolino and R. Pukhtaievych (University of Padova) which provides explicit computations of the power series expansion for the effective conductivity of a non-ideal composite. This paper puts the basis for the numerical implementations considered in the next objective iv);
v) the fellow is working on this objective with P. Musolino, G. Mishuris, and engineers such as M. Gei and L. Morini (Cardiff University);
vi) the fellow has accomplished some exploratory work on objective vi).

Moreover:

- The fellow is working on a book on FAA method with P. Musolino and M. Lanza de Cristoforis. A contract has been signed with Springer (the main editor in mathematics).

- Together with G. Mishuris the fellow is supervising the research activity of the PhD candidate R. Molinarolo at AU.

OVERVIEW

The researcher has obtained results on the objectives i)-v) which lead to the production of 5 papers + 3 other papers which are in the last stage of preparation. Of the 5 finished papers one is already appeared in a major mathematical journal (Journal of Differential Equations) and 4 have been submitted for the publication. Objective iii) has revealed to be strategic also for the achievement of objectives i), iv), and v). As a consequence iii) has received a special attention. The theoretical results obtained in ii) can be now exploited for specific applications. The researcher has accomplished some exploratory work on objective vi) and has establishing theoretical foundations for the development.
The results obtained in the frame of the project represents a novelty with respect to the state of the art for the following reasons:
• The FAA shows how to obtain fully justified asymptotic expansions in terms of analytic functions and thus of converging power series (instead of the approximation formulas that typically are used in analog problems);
• The FAA allows the treatment of boundary value problems with non-linear boundary conditions. Singular perturbation problems with such conditions are very natural in continuum mechanics but far less studied in literature than the linear ones. In this action instead, nonlinear conditions have been widely investigated.

The final goal within the action is the realisation of a complete theory based on the FAA which will represent in future a key tool for the analysis of perturbed boundary value problems.

Moreover, the actions will be an asset for the researcher and in view of future establishment in a permanent position in a European university.