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 γ, L > 0 and $1<p\le q<min\{p+\frac{1}{n},\frac{2n-1}{2n-2}p\}$. We study minimizers of the functional $F[·]$ and prove a partial $C_{\mathrm{loc}}^{m,\alpha }$-regularity result.

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 γ, L > 0 and $1<p\le q<min\{p+\frac{1}{n},\frac{2n-1}{2n-2}p\}$. We study minimizers of the functional $F[·]$ and prove a partial $C_{\mathrm{loc}}^{m,\alpha }$-regularity result.

We prove partial regularity for minimizers of the functional ${\int }_{\Omega }f(x,u\left(x\right),Du\left(x\right))dx$ where the integrand f(x,u,ξ) is quasiconvex with subquadratic growth: $\left|f(x,u,\xi )\right|\le L(1+|\xi {|}^p)$, p < 2. We also obtain the same results for ω-minimizers.

Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_0$ and $x_1$, providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is aWasserstein space of probabilities on a given set $\Omega$ and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures ${\mu }_0$ and ${\mu }_1$ by means of finite cost paths are given.

Let W be a function from the real m×n-matrices to the real numbers. If W is quasiconvex in the sense of the calculus of variations, then we show that W can be approximated locally uniformly by quasiconvex polynomials.

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration...

We characterize generalized Young measures, the so-called DiPerna–Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of ℝm × n by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, Arch. Ration. Mech. Anal. 86 (1984) 251–277]. As a consequence we get...

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 $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set ${\text{M}}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $𝔸$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.