Nonlocal Variational Problems Research Articles

A note on some non-local variational problems

We study two non-local variational problems that are characterized by the presence of a Riesz-like repulsive term that competes with an attractive term. The first functional is defined on the subsets of the Euclidean space and has the fractional perimeter as an attractive term. The second functional instead is defined on non-negative integrable and uniformly bounded densities and contains an attractive term of positive-power type. For both of the functionals, we prove that balls are the unique minimizers in the appropriate volume constraint range, generalizing the results already present in the literature for more specific energies.

On a special class of non-local variational problems

On a special class of non-local variational problems

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.

Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition.With the goal of establishing a comprehensive existence theory, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we surprisingly identify quasiconvexity, which is intrinsic to the standard vectorial calculus of variations, as the natural notion also in the fractional setting. In the absence of quasiconvexity, we determine a representation formula for the corresponding relaxed functionals, obtained via partial quasiconvexification outside the region where complementary values are prescribed. Thus, in contrast to classical results, the relaxation process induces a structural change in the functional, turning the integrand from a homogeneous into an inhomogeneous one. Our proofs rely crucially on an inherent relation between classical and fractional gradients, which we extend to Sobolev spaces, enabling us to transition between the two settings.

Minimisers of a fractional seminorm and nonlocal minimal surfaces

The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev–Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. Specifically, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the W^{s,1} -seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin–Yang result describing the full and void patterns of nonlocal minimal surfaces.

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.

Regularity analyses and approximation of nonlocal variational equality and inequality problems

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal vector calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.

Quantitative estimates for bending energies and applications to non-local variational problems

AbstractWe discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

Lower Semicontinuity and Relaxation Via Young Measures for Nonlocal Variational Problems and Applications to Peridynamics

We study nonlocal variational problems in $L^p$, like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal Poincare inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in $L^p$ in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, nonpositive integrands may ...

On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

Thresholds for Existence of Dispersion Management Solitons for General Nonlinearities

We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large noncompact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion-managed solitons is shown under very mild conditions on the dispersion profile and the nonlinear polarization of an optical active medium, which cover all physically relevant cases for the dispersion profile and a large class of nonlinear polarizations; for example, they are allowed to change sign.

Weak lower semicontinuity and relaxation for a class of non-local functionals

Motivated by a recent interest in models of peridynamics, we survey, gather, and summarize various results, that are scattered throughout the literature, about weak lower semicontinuity of some non-local variational problems in which non-locality is expressed in the form of a double integral. Weak lower semicontinuity translates into separate convexity, though one has to incorporate in such a result a most unusual phrase to stress that weak lower semicontinuity ought to be true for all domains. The most striking fact, however, is that relaxation cannot be written and understood as a functional of the same kind but something more involved needs to be found.

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

We present a nonconforming discontinuous Galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints. The nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions. The key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin. It works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions. A convergence theory is established for two types of finite dimensional spaces, with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown...

A class of nonlocal variational problems on a vector bundle for color image local contrast reduction/enhancement

We extend two existing variational models from the Euclidean space to a vector bundle over a Riemannian manifold. The Euclidean models , dedicated to regularize or enhance some color image features, are based on the concept of nonlocal gradient operator acting on a function of the Euclidean space. We then extend these models by generalizing this operator to a vector bundle over a Riemannian manifold with the help of the parallel transport map associated to some class of covariant derivatives. Through the dual formulations of the proposed models, we obtain the expressions of their solutions, which exhibit the functional spaces that describe the image features. Finally, for a well-chosen covariant derivative and its nonlocal extension, the proposed models perform local contrast modification (reduction or enhancement) and experiments show that they preserve more the aspect of the original image than the Euclidean models do while modifying equally its contrast.

Existence for Nonlocal Variational Problems in Peridynamics

We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in solid mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincaré inequality. We cover Dirichlet, Neumann, and mixed boundary conditions. The existence theory is set in the Lebesgue $L^p$ spaces and in the fractional Sobolev $W^{s,p}$ spaces, for $0 < s < 1$ and $1 < p < \infty$.

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.

Symmetry of minimizers for some nonlocal variational problems

We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the existence of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the Davey–Stewartson equation.

An Application of the Modular Function in Nonlocal Variational Problems

Using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.

On the Multiplicity of Solutions of Two Nonlocal Variational Problems

We study two nonlocal variational problems in this paper. One models microphase separation of diblock copolymers and the other models solid-solid phase transformations that lead to fine structures. We study a parameter range where the problems can be approximated by their asymptotic limits. We find all the local minimum solutions of the limiting problems. Because these local minima are isolated, and hence stable under perturbation, near them there exist local minimum solutions of the original problems.

Nonlocal variational problems arising in long wave propagatioN

In this paper we study the existence of minimizer for certain constrained variational problems given by functionals with nonlocal terms. This type of functionals are first integrals of evolution equations describing long wave propagation and the existence of minimizer gives the existence and the stability of traveling waves for these equations. Due to loss of compactness, the major problem is to prevent dichotomy of minimizing sequences. Our approach is an alternative to the concentration-compactness method and it allows us to deal with some functionals for which the verification of the strict subadditivity seems to be difficult.