The aim of this paper is to study a type of nonlocal elliptic equation whose format includes a kernel k and a design function h. We analyze how this equation is connected with the classical elliptic equation that includes h as diffusive term. On one hand, the spectrum of the nonlocal operator that defines the nonlocal equation is studied. Existence and unicity of solutions for the nonlocal equation are proved. On the other hand, the convergence of these solutions to the solution of the classical elliptic equation as the kernel k converges to a Dirac Delta is analyzed. This work is performed by using an spectral theorem on the nonlocal operator and by applying some specific compactness results. The kernel k is assumed to be radial. Dirichlet boundary conditions are assumed for the classical problem, whereas for the nonlocal equation a nonlocal boundary Dirichlet constraint must be defined.
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In this work we are going to prove the functional J defined byis weakly lower semicontinuous in W 1,p (Ω) if and only if W is separately convex. We assume that Ω is an open set in R n and W is a real-valued continuous function fulfilling standard growth and coerciveness conditions. The key to state this equivalence is a variational result established in terms of Young measures.
It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see ).
a b s t r a c tTorsion tests at high temperatures and high strain rates were conducted on a high nitrogen steel (HNS). Under these conditions, adiabatic heating influences its flow behavior. This work focus on a new algorithm for conducting the adiabatic heating correction of stress-strain curves. The algorithm obtains the stress-strain curves at quasi-isothermal conditions from those at adiabatic conditions. The corrections in stress obtained can be higher than 15% and increase with increasing strain rates and decreasing temperatures. On the other hand, an upper bound for the temperature rise was found using a dynamic material behavior approach. Finally, the influence of adiabatic heating correction on the Garofalo equation parameters of HNS was analyzed. High values of activation energy and stress exponent were attributed to reinforcement by dispersed particles and the high amount of alloying elements.
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