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.