A Variational Theory of Convolution-Type Functionals by Roberto Alicandro & Nadia Ansini & Andrea Braides & Andrey Piatnitski & Antonio Tribuzio

Author:Roberto Alicandro & Nadia Ansini & Andrea Braides & Andrey Piatnitski & Antonio Tribuzio
Language: eng
Format: epub
ISBN: 9789819906857
Publisher: Springer Nature Singapore

where the last inequality is a consequence of (2.​5) and the fact that the integration domain is far from the origin. We can choose an index 1 ≤ kε ≤ N − 4 such that

(5.13)

uniformly in T, and

Letting first N → +∞ and then T → +∞, by Lemma 5.1 we get the conclusion. □

Remark 5.1

Reasoning as in Proposition 5.1 we can also derived the following inequality for the

(5.14)

Proposition 5.2 (Inner Regularity)

Let Fε(⋅, ⋅) be defined by (2.​3) and assume that (H0)–(H2) hold. Then

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