Evolutionary problems with p,q-growth

Principal investigator: Verena Bögelein
Staff member: Thomas Singer

The project is funded by the DFG (Deutsche Forschungsgemeinschaft). Having started at the University of Erlangen-Nürnberg in 2013, it was shifted to the University of Salzburg in 2014.

Description of the project: The aim of this project is to establish an extensive understanding of evolutionary equations with coefficients satisfying a non-standard 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 and minimizers, respectively, are investigated. By now, many results have been achieved in the stationary case. On the contrary, the time-dependent parabolic case is still almost completely 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), 100(4):535-563, 2013] and [V. Bögelein, F. Duzaar, P. Marcellini. Parabolic systems with p,q-growth: a variational approach. Arch. Ration. Mech. Anal., 210(1):219-267, 2013]. In these papers, two different notions of solution were introduced: weak solutions and variational solutions (or parabolic minimizers). 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 of 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.

Publications

V. Bögelein, F. Duzaar, P. Marcellini, and S. Signoriello. Parabolic equations and the bounded slope condition, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 34(2), 355-379, 2017.
V. Bögelein, F. Duzaar, and C. Scheven. The obstacle problem for the total variation flow, Ann. Sci. Éc. Norm. Supér. 49(5), 1143-1188, 2016.
T. Singer. Local Boundedness of variational solutions to evolutionary problems with non-standard grwoth, NoDEA Nonlinear Differential Equations Appl. 23:19, 2016.
T. Singer. Existence of weak solutions of parabolic systems with p,q-growth, Manuscripta Math. 151, 87-112, 2016.
V. Bögelein, F. Duzaar and P. Marcellini. A time dependent variational approach to image restoration, SIAM J. Imaging Sci. 8(2), 968-1006, 2015.
V. Bögelein, F. Duzaar, P. Marcellini, and S. Signoriello. Nonlocal diffusion equations, J. Math. Anal. Appl. 432(1), 398-428, 2015.
T. Singer. Parabolic equations with p,q-growth: The subquadratic case, Q. J. Math. 66 (2), 707-742, 2015.