Differential and Integral Equations

Existence and regularity of minimizers for nonlocal energy functionals

Mikil D. Foss, Petronela Radu, and Cory Wright

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In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.

Article information

Differential Integral Equations, Volume 31, Number 11/12 (2018), 807-832.

First available in Project Euclid: 25 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49K21: Problems involving relations other than differential equations 49J99: None of the above, but in this section 49N60: Regularity of solutions 34B10: Nonlocal and multipoint boundary value problems 31B10: Integral representations, integral operators, integral equations methods 45G15: Systems of nonlinear integral equations


Foss, Mikil D.; Radu, Petronela; Wright, Cory. Existence and regularity of minimizers for nonlocal energy functionals. Differential Integral Equations 31 (2018), no. 11/12, 807--832. https://projecteuclid.org/euclid.die/1537840870

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