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

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

We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions ∈ ²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for (.) is proved and the asymptotic behaviour of the solutions is investigated.

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the -well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define $$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$ and let denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to ) of $$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$ Secondly the finite element approximation to the...

In this paper we study the lower semicontinuity problem for a supremal functional of the form $F(u,\Omega )=\underset{x\in \Omega }{\mathrm{ess}\sup }f(x,u\left(x\right),Du\left(x\right))$ with respect to the strong convergence in (Ω), furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for ...