Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

Abstract

In this work we consider the problemswhere L is a nonlocal differential operator and Ω is a bounded domain in R N , with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f . In the process we establish an L p -theory, for p 1, associated to these problems and we prove some useful inequalities for the applications.2010 Mathematics Subject Classification. 45K05, 47G20, 35R09, 35D30, 35D35.

Controllability of the one-dimensional fractional heat equation under positivity constraints

“…We recall that according to [20,Theorem 26], for any u ∈ L 2 ((0, T ); H −s (−1, 1)) and z 0 ∈ L 2 (−1, 1), the system (2.1) admits a unique weak solution 1)). Moreover, if u ∈ L 2 (ω × (0, T )) and z 0 ≡ 0, then it has been shown in [5, Theorem 1.5] that z ∈ L 2 ((0, T ); H 2s loc (−1, 1)) ∩ L ∞ ((0, T ); H s 0 (−1, 1)) and z t ∈ L 2 ((−1, 1) × (0, T )).…”

Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

“…Due to the nonlocality of the fractional Laplacian, several notions of regularity can be studied. The following results, which generalize the fractional regularity proved in [34,Theorem 24] with a different approach, can be seen as the counterpart of Proposition 3.1 to deal with (Q λ ) and ( Q λ ).…”

“…See for instance [37, Chapter 5]. b) In the particular case of the fractional Laplacian of order s ∈ (1/2, 1) and for h ∈ L 1 (Ω), we improve the regularity results of [1,31,34]. Note however that in the three quoted papers the authors deal with more general operators and cover the full range s ∈ (0, 1).…”

Existence of a unique positive entropy solution to a singular fractional Laplacian

“…Moreover, for every φ ∈ T , there exists a constant β ∈ (0, 1) such that φ ∈ C 0,β (Ω). See [36,35,37]. It is easy to check that for u ∈ X s 0 (Ω) and φ ∈ T :…”