By Giovanni Molica Bisci, Vicentiu D. Radulescu, Raffaella Servadei
ISBN-10: 1107111943
ISBN-13: 9781107111943
This booklet presents researchers and graduate scholars with an intensive creation to the variational research of nonlinear difficulties defined by means of nonlocal operators. The authors provide a scientific therapy of the fundamental mathematical concept and positive tools for those periods of nonlinear equations, plus their program to numerous tactics bobbing up within the technologies. The equations are tested from a number of viewpoints, with the calculus of adaptations because the unifying subject matter. half I starts off the ebook with a few easy evidence approximately fractional Sobolev areas. half II is devoted to the research of fractional elliptic difficulties regarding subcritical nonlinearities, through classical variational tools and different novel methods. ultimately, half III features a choice of contemporary effects on serious fractional equations. A cautious stability is struck among rigorous arithmetic and actual purposes, permitting readers to determine how those diversified issues relate to different vital parts, together with topology, useful research, mathematical physics, and capability idea.
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62). Proof Let u be in X 0s ( ). 4 (here with p = 2), we get u 2 ∗ L 2s (Rn ) ≤c Rn ×Rn |u(x) − u(y)|2 d x d y, |x − y|n+2s where c is a positive constant depending only on n and s. e. in C , we get assertion (a). 62) it easily follows that 2 Xs( ) u ≥ |u(x) − u(y)|2 K (x − y) d x d y. 56), we get 1/2 u 2 Xs ( ) = u L2( ) + bounded and 2 |u(x) − u(y)| K (x − y)d x d y 2 Q ≤2 u 2 +2 L2( ) ∗ |u(x) − u(y)|2 K (x − y)d x d y Q ∗ ≤ 2| |(2s −2)/2s u ∗ 2 +2 ∗ L 2s ( ) ∗ ≤ 2c| |(2s −2)/2s Rn ×Rn |u(x) − u(y)|2 K (x − y) d x d y Q |u(x) − u(y)|2 dx dy |x − y|n+2s |u(x) − u(y)|2 K (x − y) d x d y +2 Q ∗ ≤2 ∗ c| |(2s −2)/2s +1 θ |u(x) − u(y)|2 K (x − y) d x d y.
48), and this ends the proof. Different definitions of the fractional Laplacian consider different normalizing constants. 8)]). 4]). Here − denotes the classical Laplacian operator. Finally, we are able to prove the relation between the fractional Laplacian operator ( − )s and the fractional Sobolev space H s (Rn ) (see [83]). 18). Then, for any u ∈ H s (Rn ), [u]2H s (Rn ) = 2C(n, s)−1 ( − )s/2 u 2 . 4), it follows that ( − )s/2 u 2 L 2 (Rn ) = F ( − )s/2 u 2 . 17 yields F ( − )s/2 u 2 L 2 (Rn ) = |ξ |F u 2 .
E. in Rn because · X s ( ) is a norm. 87) as a norm on X 0s ( ). 29 u, v X 0s ( ) is a Hilbert space with scalar product X 0s ( ) := Rn ×Rn (u(x) − u(y))(v(x) − v(y))K (x − y) d x d y. 88) Proof First of all, let us consider the map X 0s ( ) × X 0s ( ) (u, v) → u, v X 0s ( ) . 87). In order to show that X 0s ( ) is a Hilbert space, it remains to prove that X 0s ( ) is complete with respect to the norm · X 0s ( ) . For this, let {u j } j∈N be a Cauchy sequence in X 0s ( ). 28(b). Hence, being L 2 ( ) complete, there exists u ∞ ∈ L 2 ( ) such that u j → u ∞ in 2 L ( ) as j → +∞.