Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

We consider an energy functional combining the square of the local oscillation of a one-dimensional function with a double-well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler–Lagrange equation. We also prove that, differently from the classical cases arising in ordinary differential equations, in this context the heteroclinic connections are not necessarily smooth, and not even continuous (in fact, they can be piecewise constant). Also, we show that heteroclinics are not necessarily unique up to a translation, which is also in contrast with the classical setting. Furthermore, we investigate the associated Dirichlet problem, studying existence, uniqueness and partial regularity properties, providing explicit solutions in terms of the external data and of the forcing source, and exhibiting an example of discontinuous solution.

Halfspaces minimise nonlocal perimeter: a proof via calibrations

We consider a nonlocal functional $$J_K$$ J K that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $$u:\mathbb {R}^d\rightarrow \mathbb {R}$$ u : R d → R , we define $$J_K(u)$$ J K ( u ) as the integral of weighted differences of u. The weight is encoded by a positive kernel K, possibly singular in the origin. We study the minimisation of this energy under prescribed boundary conditions, and we introduce a notion of calibration suited for this nonlocal problem. Our first result shows that the existence of a calibration is a sufficient condition for a function to be a minimiser. As an application of this criterion, we prove that halfspaces are the unique minimisers of $$J_K$$ J K in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a $$\varGamma $$ Γ -convergence result concerning the scaling limit of $$J_K$$ J K .

On One Damping Problem for a Nonstationary Control System with Aftereffect

We consider a control system described by the system of differential-difference equations of neutral type with variable matrix coefficients and several delays. We establish the relation between the variational problem for the nonlocal functional describing the multidimensional control system with delays and the corresponding boundary-value problem for the system of differential-difference equations. We prove the existence and uniqueness of the generalized solution of this boundary-value problem.

Exponential Stability of Traveling Waves for a Reaction Advection Diffusion Equation with Nonlinear-Nonlocal Functional Response

The stability of a reaction advection diffusion equation with nonlinear-nonlocal functional response is concerned. By using the technical weighted energy method and the comparison principle, the exponential stability of all noncritical traveling waves of the equation can be obtained. Moreover, we get the rates of convergence. Our results improve the previous ones. At last, we apply the stability result to some real models, such as an epidemic model and a population dynamic model.

Local and Nonlocal Regularization to Image Interpolation

This paper presents an image interpolation model with local and nonlocal regularization. A nonlocal bounded variation (BV) regularizer is formulated by an exponential function including gradient. It acts as the Perona-Malik equation. Thus our nonlocal BV regularizer possesses the properties of the anisotropic diffusion equation and nonlocal functional. The local total variation (TV) regularizer dissipates image energy along the orthogonal direction to the gradient to avoid blurring image edges. The derived model efficiently reconstructs the real image, leading to a natural interpolation which reduces blurring and staircase artifacts. We present experimental results that prove the potential and efficacy of the method.