Teaching Responsibilities 2016/17:
Term 2: MA4G6 Calculus of Variations
Most of my research concerns singularities in nonlinear PDEs and the modern theory of the calculus of variations. In particular, I am interested in oscillation and concentration phenomena and what can be rigorously proved about their "shape". Applications include elasticity and elasto-plasticity theory.
On a more technical level, I work on:
- Efficient description of oscillations and concentrations
- Lower semicontinuity and quasiconvexity
- Compensated compactness and, more generally, restrictions on oscillations and concentrations
- Fine properties of functions of bounded variation/deformation
- Rigidity and softness, convex integration
- Rate-independent systems and plasticity
I also worked on telecommunications networks and have recently become interested in anomaly detection in streaming data.
Slides from talks about several aspects of my research can be found below.
- Regularity and approximation of strong solutions to rate-independent systems (with S. Schwarzacher, E. Süli), Preprint.
- Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints (with A. Arroyo-Rabasa, G. De Philippis), Preprint.
- On a conjecture of Cheeger (with G. De Philippis, A. Marchese), to appear in "Measure Theory in Non-Smooth Spaces" (Nicola Gigli, ed.), De Gruyter, Preprint.
- Characterization of generalized Young measures generated by symmetric gradients (with G. De Philippis), to appear in Arch. Ration. Mech. Anal., Online version.
- On the structure of A-free measures and applications (with G. De Philippis), Ann. of Math. 184 (2016), 1017-1039, Online version.
- Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms. Arch. Ration. Mech. Anal. 215 (2015), 1-63. Online version.
- Differential inclusions and Young measures involving prescribed Jacobians (with K. Koumatos, E. Wiedemann), SIAM J. Math. Anal. 47 (2015), 1169-1195. Online version.
- A local proof for the characterization of Young measures generated by sequences in BV, J. Funct. Anal. 266 (2014), 6335-6371. Online version.
- Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), 63-113. Online version.
- Characterization of generalized gradient Young measures in W1,1 and BV (with J. Kristensen), Arch. Ration. Mech. Anal. 197 (2010), 539-598. Online version.
A complete list of publications can be found in my CV.
Lecture notes / theses / slides
- Introduction to the Modern Calculus of Variations, Lecture notes for a 30h fourth-year module, January 2017. Download.
- The positive Jacobian constraint in elasticity theory and orientation-preserving Young measures, Lecture notes for a short course, August 2015. Download.
- Lower Semicontinuity and Young Measures for Integral Functionals with Linear Growth, DPhil thesis, University of Oxford, November 2011. Download.
- Reverse Approximation of Rate-Independent Evolution Processes, Diploma thesis, Technical University Berlin, September 2008. Download.
- On the structure of PDE-constrained measures and applications, Slides, June 2016. Download.
- The positive Jacobian constraint in elasticity theory and orientation-preserving Young measures, Slides, January 2014. Download.
- Directional oscillations and concentrations and weak/strong compactness via microlocal compactness forms, Slides, December 2012. Download.
- Lower semicontinuity for minimization problems in the space BD of functions of bounded deformation, Slides, November 2011. Download.