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Register a new userLower semicontinuity and relaxation of nonlocal $L^infty$-functionals
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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.
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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.
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}.
In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.
The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^infty (mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power.
We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.
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