CALCULUS OF VARIATIONS: REGULARITY THEORY AND LIMITING SPACES
Titles & abstracts
Invited speakers
IRENE FONSECA -- Title: Phase Separation in Heterogeneous Media Abstract: Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).
GIUSEPPE MINGIONE: TBA
LUBOS PICK -- Title: Maximal noncompactness of Sobolev embeddings Abstract: It had been known for some time that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of weak Lebesgue spaces had been open, too. We will survey certain recently obtained general principles, which, in particular, provide answers to these and related problems.
THOMAS SCHMIDT -- Title: Perimeter functionals with measure data.
ANDREA CIANCHI --- Title: Distortion of Hausdorff measures under Orlicz-Sobolev maps Abstract. A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved. This is joint work with M.V.Korobkov and J.Kristensen.
MATHIAS SCHÄFFNER -- Title: Regularity for nonuniformly elliptic equations Abstract: I discuss regularity results for solutions of linear nonuniformly elliptic equations. If time permits, I will discuss applications in stochastic homogenization and regularity for nonuniformly elliptic variational integrals.
ANDRÉ GUERRA -- Title: Quasiconvexity and nonlinear Elasticity. Abstract: Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.
CRISTIANA DE FILIPPIS -- TBA
PAOLO BARONI -- Title: New results for non-autonomous functionals with mild phase transition Abstract: We describe how different regularity assumptions, of integral type, on the x-dependence of the energy impact the regularity of minimizers of some non-autonomous functionals having moderate nonuniform ellipticity. We put particular emphasis on double phase functionals with logarithmic phase transition but also functionals with p(x) growth; we shall include some new results.
LARS DIENING -- TBA
Contributed talks
STEFAN SCHIFFER -- Title: On truncations subject to differential constraints. Abstract: In this talk, I give an overview over recent results regarding truncation with respect to some differential constraint. Compared to previous works on Lipschitz truncation, we focus on the low-regularity case. A key question may be formulated as follows: How can a divergence-free $L^1$ function be slightly modified, such that the modification is in $L^{\infty}$ and still divergence-free? This limiting case is unsurprisingly much harder than the corresponding $L^p$-$L^q$ truncation for $1<p,q
LUKAS KOCH -- Title: A p-harmonic approximation result in optimal transportation Abstract: I will present recent progress towards deriving a partial regularity theory for optimal transportation with cost |x-y|^p. The key observation is a p-harmonic approximation result. The talk will emphasize connections to techniques in elliptic regularity as well as applications of the result to optimal matching problems.
AMIRAN GOGATISHVILI -- Title: Fractional order Orlicz-Sobolev space
ELEONORA FICOLA -- Title: Lower semicontinuity and existence results for total variation functionals with measure data
PETER LEWINTAN -- Title: L^1-Korn-Maxwell-Sobolev inequalities in all dimensions Abstract: We characterize all linear part maps A[·] (e.g. A = sym) which may appear on the right hand side of Korn-Maxwell-Sobolev inequalities for incompatible tensor fields P. The correction term Curl P appears thereby in the L^1-norm on the right hand side. Different from previous contributions, the results to be presented are applicable to all dimensions and optimal. This particularly necessitates the distinction of different constellations between the ellipticities of A and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation. These results are based on a joint work with Franz Gmeineder (Konstanz) and Patrizio Neff (Essen).