Differentiable vs non-differentiable systems

Differentiable vs non-differentiable systems

Nonautonomous, nonuniformly elliptic functionals are variational integrals characterised by quite a wild behaviour of the ellipticity ratio associated to their integrand, in the sense that it may blow up as the modulus of the gradient variable goes to infinity.

We analyse the interaction between the space-depending coefficient of the integrand and a possible forcing term and derive optimal Lipschitz criteria for minimizers. We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth. We also find new borderline regularity results also in the uniformly elliptic case, i.e. when the ellipyicity ratio is uniformly bounded.

This approach yields optimal regularity results for obstacle problems associated for instance to iterated exponential models, which have been treated in [2] for the first time. Finally, we look at general nonautonomous integrands with (p,q)-growth and show general interpolation properties allowing to get basic higher integrability results for either bounded or Hölder continuous minimizers under improved bounds for the gap q-p.

This talk is based on papers [1,2,3].

References

• [1] C. De Filippis, G. Mingione, Interpolative gap bounds for nonautonomous integrals. Preprint (2020), submitted.
• [2] C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals. Preprint (2020), submitted. https://arxiv.org/pdf/2007.07469.pdf
• [3] C. De Filippis, G. Mingione, On the regularity of minima of non-autonomous functionals. Journal of Geometric Analysis 30:1584-1626, (2020). https://doi.org/10.1007/s12220-019-00225-z

Speaker

Cristiana de Filippis, University of Oxford

How to attend

If not a member of the Dept. Mathematical Science at the University of Essex, you can register your interest in attending the seminar and request the Zoom’s meeting password by emailing Dr Jesus Martinez-Garcia jesus.martinez-garcia@essex.ac.uk.