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Register a new userA relaxation result in the vectorial setting and $L^p$-approximation for $L^infty$-functionals
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We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,infty}(Omega;mathbb R^d) i u mapstosupess_{ x in Omega}f( abla u(x))$ in the vectorial case, where $Omegasubset mathbb R^N$ is a Lipschitz, bounded open set, and $f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,xi)$, which are only $L^N otimes B_{d times N}$-measurable.
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We study variational problems involving nonlocal supremal functionals $L^infty(Omega;mathbb{R}^m) i umapsto {rm ess sup}_{(x,y)in Omegatimes Omega} W(u(x), u(y)),$ where $Omegasubset mathbb{R}^n$ is a bounded, open set and $W:mathbb{R}^mtimesmathbb{R}^mto mathbb{R}$ is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for $L^infty$-weak$^ast$ lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. Whether the same statement holds in the related context of double-integral functionals is currently still open. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak$^ast$ closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.
Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}rightarrow L^{infty }$ framework to construct global unique solutions near Maxwellian with small $L^{infty } $norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgis method cite{golse2016harnack} and cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgis method cite{golse2016harnack}, cite{golse2015holder}, and cite{mouhot2015holder}.
An optimal control problem with a time-parameter is considered. The functional to be optimized includes the maximum over time-horizon reached by a function of the state variable, and so an $L^infty$-term. In addition to the classical control function, the time at which this maximum is reached is considered as a free parameter. The problem couples the behavior of the state and the control, with this time-parameter. A change of variable is introduced to derive first and second-order optimality conditions. This allows the implementation of a Newton method. Numerical simulations are developed, for selected ordinary differential equations and a partial differential equation, which illustrate the influence of the additional parameter and the original motivation.
Avikainen showed that, for any $p,q in [1,infty)$, and any function $f$ of bounded variation in $mathbb{R}$, it holds that $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensiona
Riemannian cubics are critical points for the $L^2$ norm of acceleration of curves in Riemannian manifolds $M$. In the present paper the $L^infty$ norm replaces the $L^2$ norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are examined when $M$ is a sphere or a bi-invariant Lie group.
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