Evolutionary problems with p,q-growth

• Applicant: Professorin Dr. Verena Bögelein
• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 250001814

The aim of this project is to establish an extensive understanding of evolutionary equations with coefficients satisfying a non-standard growth condition (a so called p,q-growth condition). In the stationary case the class of partial differential equations / variational integrals with p,q-growth was discovered by P. Marcellini in the 80ies. Thereby, he found surprising examples of minimizers of variational integrals with singularities. Since then the investigation of this kind of differential equations/variational integrals is of large interest. In particular questions of existence, regularity and irregularity of solutions, respectively minimizers are investigated. By now, in the stationary case there have already many results been achieved. On the contrary, the time dependent parabolic case is still almost open. There are only few results for very special cases. The reason may be on the one hand that the parabolic case is much more involved and on the other hand that the right notion of solution had not been resolved. Recently, this question has been clarified in [V. Bögelein, F. Duzaar, P. Marcellini, Parabolic equations with p,q-growth. J. Math. Pures Appl. (9), DOI:10.1016/j.matpur.2013.01.012.] and [V. Bögelein, F. Duzaar, P. Marcellini. Parabolic systems with p,q-growth: a variational approach. Arch. Ration. Mech. Anal., DOI:10.1007/s00205-013-0646-4]. In these papers, two different notions of solution were introduced, on the one hand the notion of a weak solution and on the other hand the notion of a variational solution (or parabolic minimizer). It seems that the notion of variational solution is much more flexible, since a priori less regularity is needed to define the solution. Moreover, weaker assumptions concerning the regularity of the integrand are possible. The two mentioned papers are the starting point for a systematic investigation of parabolic problems with p,q-growth. Therefore, the goal of this project is to establish a general and comprehensive existence and regularity theory for this kind of problems. In particular, a proof of the existence of variational solutions based on purely variational methods would be on large interest and is one of the main goals of the project. Such a proof would approve that the approach via variational solutions is the natural one.

• DFG Programme: Research Grants
• International Connection: Austria