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Register a new userCoupled local/nonlocal models in thin domains
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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).
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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.
We consider the problem of optimal distribution of a limited amount of conductive material in systems governed by local and non-local scalar diffusion laws. Of particular interest for these problems is the study of the limiting case, which appears when the amount of available material is driven to zero. Such a limiting process is of both theoretical and practical interest and continues to be a subject of active study. In the local case, the limiting optimization problem is convex and has a well understood basis pursuit structure. Still this local problem is quite challenging both analytically and numerically because it is posed in the space of vector-valued Radon measures. With this in mind we focus on identifying the vanishing material limit for the corresponding nonlocal optimal design problem. Similarly to the local case, the resulting nonlocal problem is convex and has the basis pursuit structure in terms of nonlocal antisymmetric two-point fluxes. In stark contrast with the local case, the nonlocal problem admits solutions in Lebesgue spaces with mixed exponents. When the nonlocal interaction horizon is driven to zero, the ``vanishing material limit nonlocal problems provide a one-sided estimate for the corresponding local measure-valued optimal design problem. The surprising fact is that in order to transform the one-sided estimate into a true limiting process it is sufficient to disregard the antisymmetry requirement on the two point fluxes. This result relies on duality and requires generalizing some of the well known nonlocal characterizations of Sobolev spaces to the case of mixed Lebesgue exponents.
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.
We consider a 2-dimensional thin domain with order of thickness {epsilon} which presents oscillations of amplitude also {epsilon} on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and at the bottom. We study the behavior of the Laplace operator with Neumann boundary condition and obtain its asymptotic homogenized limit as the parameter {epsilon} goes to 0. We are interested in understanding how this different oscillatory behavior at the boundary, influences the limit problem.
We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. We use a well-known semigroup technique that hinges on the strong solvability of the corresponding steady-state elliptic system. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. For an operator possessing an asymmetric kernel comparable to that of the fractional Laplacian, we prove the $L^2$-solvability of the elliptic system in a Bessel potential space using the Fourier transform and textit{a priori} estimates. This $L^2$-solvability together with the Hille-Yosida theorem is used to prove the well posedness of the wave-type time dependent problem. For the fractional Laplacian kernel we extend the solvability to $L^p$ spaces using classical multiplier theorems.
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