## Displaying similar documents to “3D-2D Asymptotic Analysis for Micromagnetic Thin Films”

### Partial regularity of minimizers of higher order integrals with (, )-growth

ESAIM: Control, Optimisation and Calculus of Variations

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We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}u:{ℝ}^{n}\supset \Omega \to {ℝ}^{N},$ where the integrand $f:{⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right)\to ℝ$, m 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition ${\gamma |A|}^{p}\le f\left(A\right)\le {L\left(1+|A|}^{q}\right)\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}A\in {⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right),$ with

### Partial regularity of minimizers of higher order integrals with (, )-growth

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

We consider higher order functionals of the form $F\left[u\right]=\underset{\Omega }{\int }f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}u:{ℝ}^{n}\supset \Omega \to {ℝ}^{N},$ where the integrand $f:{⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right)\to ℝ$, m 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition ${\gamma |A|}^{p}\le f\left(A\right)\le {L\left(1+|A|}^{q}\right)\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}A\in {⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right),$ with

### Penultimate approximation for the distribution of the excesses

ESAIM: Probability and Statistics

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Let be a distribution function (d.f) in the domain of attraction of an extreme value distribution ${H}_{\gamma }$; it is well-known that , where is the d.f of the excesses over , converges, when tends to , the end-point of , to ${G}_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where ${G}_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function which verifies ${lim}_{u\to {s}_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}\left(x/\sigma \left(u\right)\right)|$ converges to faster than $d\left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\gamma }\left(x/\sigma \left(u\right)\right)|$.