"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."
