The Galerkin–Fourier method for the study of nonlocal parabolic equations

Andrés, Fuensanta; Múñoz, Julio

doi:10.1007/s00033-019-1136-9

Cited by 1 publication

(3 citation statements)

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“…All details for the derivation of these achievements can be found at papers. 30,31,38,42,43 The main result of this paper is the existence of nonlocal optimal design (Theorem 4). The proof in given in Section 3.…”

Section: Contributions and Organization Of The Papermentioning

confidence: 99%

“…Moreover, u is the weak solution of the local problem . We refer the reader to previous works, for instance, for the details concerning this kind of nonlocal convergence.…”

Section: Preliminariesmentioning

confidence: 99%

“…In addition, and for the sake of clarity, Section contains two compactness results that have been essential in our research, and they will be used again in this article (Theorems and ). All details for the derivation of these achievements can be found at papers …”

Section: Introductionmentioning

confidence: 99%

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Existence and approximation of nonlocal optimal design problems driven by parabolic equations

Andrés

Múñoz

Rosado

2019

Math Methods in App Sciences

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This work is a follow-up to a series of articles by the authors where the same topic for the elliptic case is analyzed. In this article, a class of nonlocal optimal design problem driven by parabolic equations is examined. After a review of results concerning existence and uniqueness for the state equation, a detailed formulation of the nonlocal optimal design is given. The state equation is of nonlocal parabolic type, and the associated cost functional belongs to a broad class of nonlocal integrals. In the first part of the work, a general result on the existence of nonlocal optimal design is proved. The second part is devoted to analyzing the convergence of nonlocal optimal design problems toward the corresponding classical problem of optimal design. After a slight modification of the problem, either on the cost functional or by considering a new set of admissibility, the G-convergence for the state equation and, consequently, the convergence of the nonlocal optimal design problem are proved.

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Section: Contributions and Organization Of The Papermentioning

confidence: 99%

“…Moreover, u is the weak solution of the local problem . We refer the reader to previous works, for instance, for the details concerning this kind of nonlocal convergence.…”

Section: Preliminariesmentioning

confidence: 99%