We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left(D^{m}u\right)\mathrm{d}x\text{for}u:{ℝ}^n\supset \Omega \to {ℝ}^N,$ where the integrand $f:{⨀}^m({ℝ}^n,{ℝ}^N)\to ℝ$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^p\le f\left(A\right)\le {L(1+|A|}^q)\text{for}\text{all}A\in {⨀}^m({ℝ}^n,{ℝ}^N),$$ with

-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by $${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$ and to the constraint $$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$ where is any continuous function satisfying -growth assumptions with . Partial results are also obtained in the case , under an additional assumption on .

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies $$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$ where is an open set of ${ℝ}^2$ and is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on .

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for with an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to and to the initial condition . Then, the gain function $G_{(H,p,q)}$ of given by 14.5cm $$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$ is well-defined. We call profile of for any ${𝒦}_{\infty }$-function which is of the same order of magnitude...