Classical Fatou--Julia theory studies the iterates of a single rational map. In non-autonomous dynamics, one instead studies compositions generated by a sequence of maps, so the rule may change at every step. This added flexibility allows Julia sets with geometric features that cannot occur for the iteration of one fixed rational map.

We study non-autonomous iteration of monomials on the Riemann sphere. For a sequence of maps of the form \(p_n(z)=a_n z^{d_n}\), with infinitely many degrees at least two, we give a complete classification of the associated Julia set. The main point is that the full two-dimensional geometry is determined by the accumulation behavior of one real sequence obtained from the coefficients and degrees. As a result, the Julia set is a union of circles, possibly including the points \(0\) and \(\infty\), and its radial structure can be prescribed explicitly.

This classification gives a simple construction principle. By choosing the coefficient sequence appropriately, one can build non-autonomous Julia sets with prescribed radial geometry. The resulting examples include Cantor families of circles, Julia sets with non-empty interior that are not the whole Riemann sphere, Julia sets with empty interior but positive planar area, and perfect Julia sets that are not uniformly perfect. These phenomena show that non-autonomous polynomial dynamics is substantially more flexible than the autonomous theory. We also explain how the monomial construction can be transported by inserting an initial rational map. This produces Julia sets described by level sets of \(|R|\), where \(R\) is a rational map. In this way, the radial examples give rise to families of lemniscates, Cassini ovals, Apollonius circles, and thick level-set bands.

It is a classical result that sufficiently regular sets can be continuously deformed into a ball while simultaneously decreasing surface area. During this deformation, however, it is surprisingly difficult to ensure that the surface area changes continuously. We present a construction that guarantees the existence of such a deformations with continuously decreasing surface area. This result can be useful in certain constructions in the calculus of variations.

We establish a characterization of the bulk energy density arising in the relaxation in $\mathrm{BV}(\Omega; \mathbb{R}^N)$ of integral functionals of the form
$$
\mathcal{F}[u;\Omega] := \int_{\Omega} F(\nabla u) dx
$$
under classical pointwise $(1,q)-$growth assumptions in the subcritical regime $1 \leq q < \frac{n}{n-1}$.

The integrand $F\in C^{\infty}\left(\mathbb{R}^{N\times n}\right)$ is assumed to satisfy
$$
\frac{1}{c}|z| \le F(z) \le c(1+|z|^{q})
$$
for some constant $c>0$.

Our results extend the relaxation theory beyond the exponent range treated in the foundational work of Bouchitté, Fonseca, and Malý \cite{MR1632814}. In particular, we obtain a complete description of the bulk contribution in the full subcritical regime, including the linear-growth case $p=1$. The analysis is based on a refined blow-up procedure and measure-representation formulas adapted to the linear growth setting. This provides a unified structural characterization of the relaxed functional within the quasiconvex framework.