## Cooperation project of the Goethe-University Frankfurt and the African Institute for Mathematical Sciences (AIMS)

### Funding programme of the German Academic Exchange Service (DAAD)

Geometric variational problems arise in various applications e.g. in physics and engineering. Classical examples are the shape of solid bodies minimizing air resistance, area minimizing surfaces with prescribed boundary properties or the shape of membranes giving rise to the lowest fundamental mode. In current mathematical language, these and other geometric variational problems can be formulated as optimization problems for domain dependent functionals on given classes of subdomains of Riemannian manifolds with our without boundary. In the context of applications, isochoric (volume preserving) or isoperimetric (perimeter preserving) constraints play a fundamental role, and they used to define the relevant classes of domains among which the optimization is performed. A typical example in this context are capillary membranes separating two fluids in a container. In mathematical terms, these objects are found as constant mean curvature (CMC) hypersurfaces with free boundary contained in the boundary of the ambient manifold, and they arise as critical points of an associated reduced volume functional. Another example of high relevance are eigenvalue optimization problems for the Laplace-Beltrami operator in subdomains of Riemannian manifolds with fixed volume constraints and the associated overdetermined boundary value problems.

As the well known link between the classical isoperimetric inequality and Sobolev’s inequality indicates, geometric variational problems are closely connected to the study of best constants in functional inequalities and corresponding extremal or almost extremal functions. Equally challenging – due to their nonlocal nature – are optimization problems for trace terms

in manifolds with boundary. In some special cases these problems reduce to nonlinear boundary value problems involving fractional powers of the Laplacian. However, even the latter problem class, which has drawn more and more attention in recent years, is far from understood. A thorough investigation of phenomena related to fractional boundary value problems is also of independent interest since the fractional Laplacian also arises in models for anomalous diffusion processes as observed in the

spread of epidemics.

The cooperation project is devoted to these and related topics in the context of partial differential equations. The aim is to support collaboration between scientists from AIMS Senegal and the Institute of Mathematics of the Goethe University Frankfurt by including doctoral candidates, postdocs and qualified master students. Moreover, we wish to increase the awareness for the newly built AIMS center in Senegal among colleagues and junior scientists from Frankfurt and other German universities. Finally, we also wish to use the cooperation between the AIMS center and the Goethe-University as a brigde which enables networking and cooperation between German and Senegalese Universities.

**Research stays of faculty members of Senegalese universities at German universities**

Faculty members of Senegalese Universities associated with AIMS-Senegal are invited to apply for a grant for visiting a German University for a period of up to one month **in 2016** for the purpose of

− initiating scientific cooperation in mathematics, and/or

− participation in mathematical conferences.

Preference will be given to applications related to Geometric Analysis and Partial Differential Equations.

The grant funds travel expenses up to 725 € and local expenses including accommodation up to 2000 € depending on the length of the visit.

Applications will be invited including

− a motivational letter including a short research statement where the aims of the stay are outlined,

− a CV of the applicant, and

− an invitation letter by a professor in mathematics of a German university.

**The deadline for applications is March 31, 2016. **

To apply, please send the documents listed above in pdf-format (preferrably one file)

habash@math.uni-frankfurt.de and weth@math.uni-franfurt.de

**Schools at AIMS Senegal for Junior Scientists**

### Spring School on Nonlinear PDEs and Related Problems, February 15 - 19, 2016

### Spring School on Variational and Geometric Methods in Nonlinear PDEs, February 15-20, 2015

### Fall School on PDEs and Probability, November 21 - 25, 2016

The programme is funded by