We study a linear singular first-order integro-differential Cauchy problems in Banach spaces. Singular here means that the integro differential equation is not in normal form neither can it be reduced to such a form. We generalize to this context some existence and uniqueness theorems known for differential equations. Particular attention is given to single out the optimal regularity properties of solutions as well as to point out several explicit applications related to singular partial integro-differential of parabolic type.

We prove the existence and uniqueness of a $C^2$--solution of the Timoshenko system for the motion of an elastic beam. In addition, we give pointwise estimates for the displacement, rotation, shear angle and their derivatives with constants explicitly calculated. The method of proof is based on the Hille--Yosida operator theory.

We consider a transmission problem for a system of electromagneto-elasticity having piecewise constant coefficients in a bounded domain. Under suitable geometric conditions imposed on the domain and the interfaces where the coefficients have a jump discontinuity, results on uniform boundary stabilization are established. Exact boundary controllability is then obtained through Russell's "controllability via stabilizability" principle.

We are interested in the effect of a constant electric field on the electronic structure of a crystal. We model it by a density-functional theory, and derive from this microscopic model macroscopic features of the system by letting the ratio $\varepsilon $ between atomic spacing and the size of the crystal go to zero. Although many aspects are disregarded in this approach, we show that the effect of the electric field is negligible inside the crystal and estimate its vanishing rate with respect to the distance from the boundary of the crystal.

We consider the equation $-\Delta_{p}u=u^{\alpha}+u^{q}$ where $0\le q <p-1 <\alpha\le p^{*}-1$ in the ball $B_{R}(0)\subset \mathbb R^{N}, N\ge 2.$ Here, $p^{*}=Np/(N-p)$. We show the existence of at least two positive solutions to the above equation for small enough balls when $\alpha=p^{*}-1$ and $q>0.$ Further if $p\in (1,2)$ and $\alpha\le p^{*}-1$, we show the existence of exactly two positive solutions for small enough balls when $q>0$, and at most two solutions when $q=0$. This we do by the asymptotic analysis of the corresponding Emden-Fowler equation.