Elisa Davoli

Elisa Davoli
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Mathematics - Analysis of PDEs (10)

Publications Authored By Elisa Davoli

A representation formula for the relaxation of integral energies $$(u,v)\mapsto\int_{\Omega} f(x,u(x),v(x))\,dx,$$ is obtained, where $f$ satisfies $p$-growth assumptions, $1Read More

A new fractional order seminorm, $ICTV^r$, $r\in \mathbb R$, $r\geq 1$, is proposed in the one-dimensional setting, as a generalization of the standard $ICTV^k$-seminorms, $k\in\mathbb{N}$. The fractional $ICTV^r$-seminorms are shown to be intermediate between the standard $ICTV^k$-seminorms of integer order. A bilevel learning scheme is proposed, where under a box constraint a simultaneous optimization with respect to the parameter $\alpha$ and the order $r$ is performed. Read More

A homogenization result for a family of oscillating integral energies is presented, where the fields under consideration are subjected to first order linear differential constraints depending on the space variable x. The work is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of A-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result. Read More

An analytical validation is obtained for the evolution equation $$h_t=\Delta[ \mathcal{F}^{-1}(-aE \mathcal{F}(h)) - r/h^2 -\Delta h ],$$ introduced in {\cite{TS}} by W.T. Tekalign and B. Read More

A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of A -quasiconvexity with variable coefficients and on two- scale convergence techniques. Read More

The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff's nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter $h$ and the two homogenization scales $\ep$ and $\ep^2$. Read More

In this paper we deduce by {\Gamma}-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by {\epsilon} the thickness of the plate, we study the case where the scaling factor of the elasto- plastic energy is of order {\epsilon}^ (2{\alpha}-2), with {\alpha}>=3. We show that solutions to the three- dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on {\alpha}. Read More

The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of {\Gamma}-convergence, in the framework of finite plasticity. Denoting by {\epsilon} the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy is of order {\epsilon}^(2{\alpha}-2), with {\alpha}>=3. According to the value of {\alpha}, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Karman plate theory and the linearized plate theory. Read More

The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and {\delta}_h, respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order {\epsilon}_h^2, with {\epsilon}_h/{\delta}_h^2 \rightarrow l \in [0, +\infty). Different linearized models are deduced according to the relative order of magnitude of {\delta}_h with respect to h. Read More

The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von K\'arm\'an plate theory for rods, and the linear theory. Read More