Nonlocal Variational Problem Research Articles

Local Hölder regularity of minimizers for nonlocal variational problems

We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) [Formula: see text]-Hölder continuous for some [Formula: see text], where [Formula: see text] is a parameter related to the nonlocal operator, we prove that the solution is also [Formula: see text]-Hölder continuous.

Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.

Relaxation of a scalar nonlocal variational problem with a double-well potential

We consider nonlocal variational problems in $$L^p$$, like those that appear in peridynamics, where the functional object of the study is given by a double integral. It is known that convexity of the integrand implies the lower semicontinuity of the functional in the weak topology of $$L^p$$. If the integrand is not convex, a usual approach is to compute the relaxation, which is the lower semicontinuous envelope in the weak topology. In this paper we compute such a relaxation for a scalar problem with a double-well integrand. The relaxation is non-trivial, and, contrary to the local case, it cannot be represented as a double integral, as the original problem. Nonetheless, we show that, as for the local case, the relaxation can be expressed in terms of the energy of a suitable truncation of the considered function.

Minimal lamellar structures in a periodic FitzHugh–Nagumo system

A singular limit of a FitzHugh–Nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions. We study the stationary lamellar set and give a criterion to select out the one with the lowest energy. Such an optimal structure is called a minimal lamella. While the empty set or the full torus is a global minimizer for appropriate parameter regimes, the minimal lamellae beat both in other circumstances. The concept of minimal lamella points out that a preferred 1D mesh size is universal.

Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

For \begin{document}$ -n \begin{document}$ 0 and \begin{document}$ K(x) = \frac{\|x\|^q}{q} - \frac{\|x\|^p}{p}, $\end{document} the existence of minimizers of \begin{document}$ E(u) = \int_{R^n\times R^n} K(x-y) u(x)u(y) \,dx \,dy $\end{document} under \begin{document}$ \int_{R^n}u(x) \, dx = m>0; \quad 0 \leq u(x) \leq M, $\end{document} with given \begin{document}$ m $\end{document} and \begin{document}$ M $\end{document} , is proved in [ 3 ]. Moreover, except for translation, uniqueness and radial symmetry of the minimizer is proved for \begin{document}$ -n and \begin{document}$ q = 2 $\end{document} . Here in the present paper, we show that, except for translation, uniqueness and radial symmetry of the minimizer hold for \begin{document}$ -n and \begin{document}$ 2\leq q \leq 4. $\end{document} Applications are given.

Continuum Limits of Nonlocal $p$-Laplacian Variational Problems on Graphs

In this paper, we study a nonlocal variational problem which consists of minimizing in $L^2$ the sum of a quadratic data fidelity and a regularization term corresponding to the $L^p$-norm of the no...

Solving the backward problem in Riesz–Feller fractional diffusion by a new nonlocal regularization method

In this paper, we consider the backward problem introduced in [48] for Riesz–Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the Lp estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introducedJ(φ)=12‖u(φ(x);x,T)−fδ(x)‖2+β2‖[Pα1⁎φ](x)‖2, where β∈(0,1) is a regularization parameter, “⁎” denotes the convolution operation and Pα1(x) is called a convolution kernel with parameter α1, which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information.

Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System

The FitzHugh--Nagumo system gives rise to a nonlocal geometric variational problem defined on subsets of a domain. The energy of a subset contains three terms: its perimeter, its volume, and a long...

Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities

We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities; they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold λcr such that minimizers for this variational problem exist if their power is bigger than λcr and no minimizers exist with power less than the critical threshold. We also give simple criteria for the finiteness and strict positivity of the critical threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions’ concentration compactness argument. Furthermore, we give precise quantitative lower bounds on the exponential decay rate of the diffraction management solitons, which confirm the physical heuristic prediction for the asymptotic decay rate. Moreover, for ground state solutions, these bounds give a quantitative lower bound for the divergence of the exponential decay rate in the limit of vanishing average diffraction. For zero average diffraction, we prove quantitative bounds which show that the solitons decay much faster than exponentially. Our results considerably extend and strengthen the results of Hundertmark and Lee [J. Nonlinear Sci. 22, 1–38 (2012) and Commun. Math. Phys. 309(1), 1–21 (2012)].

A stationary core-shell assembly in a ternary inhibitory system

A ternary inhibitory system motivated by the triblock copolymer theoryis studied as a nonlocal geometric variational problem. The free energyof the system is the sum of two terms: the total size of the interfacesseparating the three constituents, and a longer ranging interaction energythat inhibits micro-domains from unlimited growth. In a particular parameterrange there is an assembly of many core-shells that exists as a stationaryset of the free energy functional. The cores form regions occupied by thefirst constituent of the ternary system, the shells form regionsoccupied by the second constituent, and the background is taken by thethird constituent. The constructive proof of the existence theorem revealsmuch information about the core-shell stationary assembly: asymptoticallyone can determine the sizes and locations of all the core-shells in theassembly. The proof also implies a kind of stability for the stationaryassembly.

Uniqueness of one-dimensional Néel wall profiles.

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin-film micromagnetic model, we analyse the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the Néel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behaviour at infinity. Thus, we establish uniqueness of the one-dimensional monotone Néel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.

The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc

The nonlocal geometric variational problem derived from the Ohta–Kawasaki diblock copolymer theory is an inhibitory system with self-organizing properties. The free energy, defined on subsets of a prescribed measure in a domain, is a sum of a local perimeter functional and a nonlocal energy given by the Green’s function of Poisson’s equation on the domain with the Neumann boundary condition. The system has the property of preventing a disc from drifting towards the domain boundary. This raises the question of whether a stationary set may have its interface touch the domain boundary. It is proved that a small, perturbed half disc exists as a stable stationary set, where the circular part of its boundary is inside the domain, as the interface, and the almost flat part of its boundary coincides with part of the domain boundary. The location of the half disc depends on two quantities: the curvature of the domain boundary, and a remnant of the Green’s function after one removes the fundamental solution and a reflection of the fundamental solution. This reflection is defined with respect to any sufficiently smooth domain boundary. It is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion. When the nonlocal energy is weighted less against the local energy, the stationary half disc sits near a maximum of the curvature; when the nonlocal energy is weighted more, the half disc appears near a minimum of the remnant function. There is also an intermediate case where the half disc is near a minimum of a combination of the curvature and the remnant function.

One-dimensional Néel walls under applied external fields

We present a detailed analysis of one-dimensional Néel walls in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Within the reduced one-dimensional thin film model, we formulate a non-local variational problem whose minimizers are given by one-dimensional Néel wall profiles. We prove existence, uniqueness (up to translations and reflections), regularity, strict monotonicity and the precise asymptotics of the decay of the minimizers in the considered variational problem.

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R2→S2 that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S1-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg–Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S2-vector fields by S1-vector fields away from the vortex balls.

Uniform energy and density distribution: diblock copolymers’ functional

We study a nonlocal variational problem arising in diblock copolymers models, whose energy is given by the Cahn–Hilliard functional plus a long-range interaction term. We prove that minimizers develop uniform energy and density distributions, thus justifying partially the highly regular microphase separation observed in diblock copolymers’ melts. We also give a new proof of the scaling law for the minimum energy. This work extends the techniques introduced in [1] where analogous results are proved for the sharp interface limit of the functional considered.

Uniform energy distribution for an isoperimetric problem with long-range interactions

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale. Moreover, we also prove an L ∞ L^\infty bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size R R fluctuates around the system average m m . The bound on the potential yields a rate of decay of these fluctuations as R R tends to + ∞ +\infty . This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation length.

Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers

We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by φ(u) = 1/2 |u| 2 H 1 - ∫ 1 0 ∫ R 2 |e itΔ u| 4 dxdt, and relying on the regularizing properties of the solution e itΔ to the free Schrodinger equation, it will be shown that φ has infinitely many critical points.

Scaling Laws in Microphase Separation of Diblock Copolymers

In this note, we study a nonlocal variational problem modeling microphase separation of diblock copolymers ([22], [3], [21]). We apply certain new tools developed in [5] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of Ohnishi et al. [23] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem.

Bounds on the micromagnetic energy of a uniaxial ferromagnet

We identify the optimal scaling law for a nonconvex, nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet. Our analysis is restricted to a certain parameter regime, in which the surface tension is sufficiently small relative to the other parameters of the problem. © 1998 John Wiley & Sons, Inc.

Young measures in a nonlocal phase transition problem

A nonlocal variational problem modelling phase transitions is studied in the framework of Young measures. The existence of global minimisers among functions with internal layers on an infinite tube is proved by combining a weak convergence result for Young measures and the principle of concentration-compactness. The regularity of such global minimisers is discussed, and the nonlocal variational problem is also considered on asymptotic tubes.