### A class of self-concordant functions on Riemannian manifolds.

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The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2the functional is ${I}_{}\left(u\right)=\frac{1}{2}{\int}_{\Omega}^{-1}{|1-|Du|}^{2}{|}^{2}+{\left|{D}^{2}u\right|}^{2}\mathrm{d}z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in ${W}_{0}^{2,2}\left(\Omega \right)$W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl....

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is ${\mathit{I}}_{\mathit{\u03f5}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int}}}_{\mathit{\Omega}}{\mathit{\u03f5}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{\u03f5}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where u belongs to the subset of functions in ${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega}\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal ...

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper...

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case ${}^{\text{'}}(\mathbb{R}\u207f,\mathbb{R}){\to}^{\nabla}\text{'}(\mathbb{R}\u207f,\mathbb{R}\u207f){\to}^{curl}\text{'}(\mathbb{R}\u207f,{\mathbb{R}}^{n\times n})$.

In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set ${\mathcal{U}}_{\mathrm{a}d}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by ${\mathcal{U}}_{\mathrm{a}d}$ and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity....

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding manifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold, defined by Rabier and Rheinboldt for determined implicit differential equations,...

We prove a ${C}^{k,\alpha}$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int}_{\Omega}f\left({D}^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand f satisfies (p,q) growth conditions.

We consider higher order functionals of the form$F\left[u\right]=\underset{\Omega}{\int}f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}u:{\mathbb{R}}^{n}\supset \Omega \to {\mathbb{R}}^{N},$ where the integrand $f:{\textstyle {\u2a00}^{m}}({\mathbb{R}}^{n},{\mathbb{R}}^{N})\to \mathbb{R}$, m≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition $${\gamma \left|A\right|}^{p}\le f\left(A\right)\le {L(1+|A|}^{q})\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}A\in {\textstyle {\u2a00}^{m}}({\mathbb{R}}^{n},{\mathbb{R}}^{N}),$$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 $F[\xb7]$ and prove a partial ${C}_{\mathrm{loc}}^{m,\alpha}$-regularity result.

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set...

Simulation completed with perturbation analysis provides a new approach for the optimal control of queuing network type systems. The objective of this paper is to calculate the sensitivity range of finite zero-order perturbation, that is, to determine the maximum and minimum size of perturbation within which zero-order propagation rules can be applied. By the introduction of the concept of virtual queue and first and second level no-input and full-output matrices, an algorithm is provided which...