On the genesis of directional friction through bristle-like mediating elements

We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl-Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry

Two explorations in Dynamical Systems and Mechanics: avoiding cones conditions and higher dimensional twist. Directional friction in bio-inspired locomotion

This thesis contains the work done by Paolo Gidoni during the doctorate programme in Matematical Analysis at SISSA, under the supervision of A. Fonda and A. DeSimone. The thesis is composed of two parts: "Avoiding cones conditions and higher dimensional twist" and "Directional friction in bio-inspired locomotion"

Existence and regularity of solutions for an evolution model of perfectly plastic plates

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable

An avoiding cones condition for the Poincar\ue9\u2013Birkhoff Theorem

We provide a geometric assumption which unifies and generalizes the conditions proposed by Fonda and Urena in [11, 12], so to obtain a higher dimensional version of the Poincar\ue9\u2013Birkhoff fixed point Theorem for Poincar\ue9 maps of Hamiltonian systems

Generalizing the Poincar\ue9\u2013Miranda theorem: the avoiding cones condition

After proposing a variant of the Poincare\u301\u2013Bohl theorem, we extend the Poincare\u301\u2013 Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \ub11. An illustrative application is provided for the study of functionals having degenerate multi-saddle points

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem p1''+λ1w1(x,p2)f1(p1)=0, in Ω, p2''+λ2w2(x,p1)f2(p2)=0,in Ω, p1'=p2'=0,on ∂Ω, where the coupling-weights w_i are sign-changing in the first variable, and the nonlinearities f_i:[0,1]→[0,+∞[ satisfy f_i(0)=f_i(1)=0, f_i(s)>0 for all s∈]0,1[, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of 2^N positive fully nontrivial solutions when the real positive parameters λ1 and λ2 are sufficiently large