Davide Donati Mathematics · TUM

Calculus of variations

Research

  1. Free-discontinuity problems

    I have studied several problems in the setting of free discontinuity functionals, including problems related to the modelling of cohesive fracture and structured deformations. My work in this area has been mainly devoted to study of the relaxation, the related question of integral representation, the asymptotic behaviour of functionals defined over spaces of functions with essential discontinuities.

    • Free discontinuity
    • Cohesive fracture
    • Structured deformations
    Structured deformations for energies with general surface termsHomogenization of vectorial free-discontinuity functionals with cohesive type surface terms Γ-convergence and homogenisation for free discontinuity functionals with linear growth in the space of functions with bounded deformation

  2. Functions of bounded variation and deformation

    Motivated by their applications to the study of free discontinuity problems in the setting of cohesive fracture, I have been studying the fine properties of functions of generalised bounded variation/deformation. In particular, in a joint work with Dal Maso we have introduced a measure which generalises the notion of symmetric gradient for functions in GBD.

    • BV and BD
    • GBV and GBD
    • Fine properties
    A new space of generalised vector-valued functions of bounded variationA matrix-valued measure associated to the derivatives of a function of generalised bounded deformationA remark on the Korn–Poincaré–Sobolev inequality for functions of generalised bounded deformation

  3. Higher-order phase transitions

    In a series of joint works with Brusca, Solci, and Trifone, I have studied extensions of the celebrated Γ-convergence result of Modica and Mortola, in which the singular perturbation is allowed to involve derivatives of any fixed, arbitrarily high, order of derivation.

    • Singular perturbations
    • Higher-order energies
    Higher-order singular perturbation models for phase transitionsSingular perturbations models in phase transitions for anisotropic higher-order materials

  4. Nonlocal models

    I have studied, via Γ-convergence, the passage from nonlocal to local models for two different classes of functionals, living in the fractional setting and in the setting of convolution-type functionals.

    • Homogenisation
    • Nonlocal-to-local limits
    • Γ-convergence
    Another look at elliptic homogenizationΓ-convergence of convolution-type functionals for free discontinuity problems
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