Peter Bella

Variational perspective on wrinkling of thin elastic sheets

Thin elastic materials can sustain much less compression than stretching - a good example being sheet of paper, which under slight confinement deforms out of plane. Often such deformations might be periodic with small length scale - we call this wrinkling. Our viewpoint is variational - we study energy of the system. The elastic energy of the sheet consists of a non-convex membrane term plus a singular perturbation (the bending term). Since the energy might have many critical points (local minima), we focus on the global minimum (ground state). The question where the wrinkles appear is well-understood, by considering relaxed variational problem (it goes under the name tension-field theory in the mechanics community). To understand properties of the wrinkling patterns (like wrinkling period and amplitude), we study expansion of the energy to the next order (the parameter being thickness of the sheet).

Publications: [6, 8, 11, 14, 17]

Stochastic homogenization

Homogenization - the computation of effective properties of materials which are heterogeneous on small scales - has been a very influential field of analysis during the last decades. The predictions of classical homogenization, however, are often restricted to periodically varying materials. Unfortunately, materials occuring in nature are almost never perfectly periodic; rather, the microscopic material properties are expected to be somewhat randomly distributed and the correlations of the microscopic properties of the material are expected to decay on larger scales. It is a basic observation of stochastic homogenization that nevertheless a homogenization result holds in many cases. Despite the qualitative theory for stochastic homogenization of e.g. elliptic equations having been settled decades ago, the quantitative theory is a very active area of research. In a recent joint work with Benjamin Fehrman, Julian Fischer, and Felix Otto, for linear elliptic PDEs and fast decorrelation of the underlying random field we establish higher-order error estimates for the homogenization error in weak spatial norms: While in Lp-type norms the homogenization error is of the order ε, for the H-1 norm of the error we derive improved bounds like ε3/2 or ε2, depending on the dimension. An earlier joint work with Benjamin Fehrman and Felix Otto is dedicated to a C1,α large-scale regularity theory and associated Liouville principles for random non-uniformly elliptic operators; here we require stationarity and ergodicity of the probability distribution of the coefficient field, together with moment bounds on the coefficient field and its inverse.

Publications: [13, 16, 18, 19, 20]