Teaching Responsibilities 2016/17:
Term 2: MA4G6 Calculus of Variations
Research workshop: "Recent Advances in PDEs and the Calculus of Variations", 3 July to 6 July 2017 in the Palazzo Pesaro-Papafava in Venice, Italy. Workshop website.
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.
Recent publications and preprints
- 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), Arch. Ration. Mech. Anal. 224 (2017) 1087-1125, 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
- 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.