Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

/pdf/hitchhiker-s-guide-to-the-fractional-sobolev-spaces-irm2sao95g.pdf

Hitchhiker's guide to the fractional Sobolev spaces

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

Mountain Pass solutions for non-local elliptic operators

In this paper we study the existence of non-trivial solutions for
equations driven by a non-local integrodifferential
operator $\mathcal L_K$ with homogeneous Dirichlet boundary
conditions. More precisely, we consider the problem
$$ \left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\
u=0  in \mathbb{R}^n \backslash Ω ,
\end{array} \right.
$$
where $\lambda$ is a real parameter and the nonlinear term $f$
satisfies superlinear and subcritical growth conditions at zero and
at infinity. This equation has a variational nature, and so its
solutions can be found as critical points of the energy functional
$\mathcal J_\lambda$ associated to the problem. Here we get such
critical points using both the Mountain Pass Theorem and the Linking
Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq
\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the
operator $-\mathcal L_K$.

As a particular case, we derive an existence theorem for the
following equation driven by the fractional Laplacian
$$ \left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) in Ω \\
u=0  in \mathbb{R}^n \backslash Ω.
\end{array} \right.
$$
Thus, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.

Variational methods for non-local operators of elliptic type

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation { (−Δ)su− λu = |u|2−2u in Ω, u = 0 in Rn \ Ω , where (−Δ)s is the fractional Laplace operator, s ∈ (0, 1), Ω is an open bounded set of Rn, n > 2s, with Lipschitz boundary, λ > 0 is a real parameter and 2∗ = 2n/(n− 2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation { LKu+ λu+ |u|2 −2u+ f(x, u) = 0 in Ω, u = 0 in Rn \ Ω , where LK is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2−2u. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if λ1,s is the first eigenvalue of the non-local operator (−Δ)s with homogeneous Dirichlet boundary datum, then for any λ ∈ (0, λ1,s) there exists a non-trivial solution of the above model equation, provided n 4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.

https://www.ams.org/tran/2015-367-01/S0002-9947-2014-05884-4/S0002-9947-2014-05884-4.pdf

Enrico Valdinoci

Papers

Hitchhiker's guide to the fractional Sobolev spaces

Hitchhiker's guide to the fractional Sobolev spaces

Mountain Pass solutions for non-local elliptic operators

Variational methods for non-local operators of elliptic type

The Brezis-Nirenberg result for the fractional Laplacian