We introduce O-systems (Definition 3.1) of orthogonal transformations of ${ℝ}^m$, and establish $ℝ$ correspondences both between equivalence classes of Clifford systems and those of O-systems, and between O-systems and orthogonal multiplications of the form $\mu :{ℝ}^n\times {ℝ}^m\to {ℝ}^m$, which allow us to solve the existence problems both for $O$-systems and for umbilical quadratic harmonic morphisms simultaneously. The existence problem for general quadratic harmonic morphisms is then solved by the Splitting Lemma. We also study properties...

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational...

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals...

Let $(M,J^{\alpha },\alpha =1,2,3)$ and $(N,{𝒥}^{\alpha },\alpha =1,2,3)$ be hyperkähler manifolds. We study stationary quaternionic maps between $M$ and $N$. We first show that if there are no holomorphic 2-spheres in the target then any sequence of stationary quaternionic maps with bounded energy subconverges to a stationary quaternionic map strongly in $W^{1,2}(M,N)$. We then find that certain tangent maps of quaternionic maps give rise to an interesting minimal 2-sphere. At last we construct a stationary quaternionic map with a codimension-3 singular set by using the embedded...