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

We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes.

http://mnikolova.perso.math.cnrs.fr/ChanEseNikoSiap06.pdf

Algorithms for finding global minimizers of image segmentation and denoising models

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

Studies in Advanced Mathematics

/pdf/a-critical-fractional-equation-with-concave-convex-power-yvlwwj6l14.pdf

A critical fractional equation with concave-convex power nonlinearities

In this work we consider the problems
$$
\left\{\begin{array}{rcll}
\mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\
u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,
\end{array}
\right.
$$
and
$$
\left\{\begin{array}{rcll}
u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\
u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\
u(x,0)&=&0 &\hbox{ in } \Omega,
\end{array}
\right.
$$
where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary.
 &nbsp
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 $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.

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

We study a porous medium equation with nonlocal diusion eects given by an inverse fractional Laplacian operator. More precisely,

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=15&iss=5&rank=4

Regularity of solutions of the fractional porous medium flow

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.

/pdf/harmonic-analysis-and-partial-differential-equations-2j4sm5v0md.pdf

Fernando Soria

Papers

Studies in Advanced Mathematics

A critical fractional equation with concave-convex power nonlinearities

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

Regularity of solutions of the fractional porous medium flow

Harmonic Analysis and Partial Differential Equations