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Register a new userLocal and nonlocal energy-based coupling models
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In this paper we study two different ways of coupling a local operator with a nonlocal one in such a way that the resulting equation is related to an energy functional. In the first strategy the coupling is given via source terms in the equation and in the second one a flux condition in the local part appears. For both models we prove existence and uniqueness of a solution that is obtained via direct minimization of the related energy functional. In the second part of this paper we extend these ideas to deal with local/nonlocal elasticity models in which we couple classical local elasticity with nonlocal peridynamics.
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In this paper, we analyze a model composed by coupled local and nonlocal diffusion equations acting in different subdomains. We consider the limit case when one of the subdomains is thin in one direction (it is concentrated to a domain of smaller dimension) and as a limit problem we obtain coupling between local and nonlocal equations acting in domains of different dimension. We find existence and uniqueness of solutions and we prove several qualitative properties (like conservation of mass and convergence to the mean value of the initial condition as time goes to infinity).
Given $Lgeq 1$, we discuss the problem of determining the highest $alpha=alpha(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^alpha_{rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $alpha(L)gtrsim {rm exp}(-CL^beta)$, for some $C, betageq 1$ depending on the dimension $Ngeq 3$. We show that in the non-local case, $alpha(L)gtrsim L^{-1-delta}$ for all $delta>0$.
In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_alpha$ defined on probability measures in $R^n$, with $ngeq 3$. The energy $I_alpha$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $alpha=0$ and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $alphain (-1, n-2]$, the minimiser of $I_alpha$ is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $n=2$, does not occur in higher dimension at the value $alpha=n-2$ corresponding to the sign change of the Fourier transform of the interaction potential.
We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and nonlocal dispersionless equations as well.
For a given Lipschitz domain $Omega$, it is a classical result that the trace space of $W^{1,p}(Omega)$ is $W^{1-1/p,p}(partialOmega)$, namely any $W^{1,p}(Omega)$ function has a well-defined $W^{1-1/p,p}(partialOmega)$ trace on its codimension-1 boundary $partialOmega$ and any $W^{1-1/p,p}(partialOmega)$ function on $partialOmega$ can be extended to a $W^{1,p}(Omega)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $mathbb{R}^dbackslashOmega$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical $W^{1-1/p,p}(partialOmega)$ space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.
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