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

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Regularization Of Inverse Problems

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Advances in mathematics

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

In this short note we prove the logarithmic stability of the single measurement uniqueness result for the fractional Calder\'on problem which had been derived in \cite{GRSU18}. To this end, we use the quantitative uniqueness results established in \cite{RS20a} and complement these bounds with a boundary doubling estimate. The latter yields control of the order of vanishing of solutions to the fractional Schr\"odinger equation. Then, following a scheme introduced in \cite{S10,ASV13} in the context of the determination of a surface impedance from far field measurements, this allows us to deduce logarithmic stability of the potential $q$.

On Single Measurement Stability for the Fractional Calder\'on Problem.

In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators, c.f. [DSV14], [DSV16].

Quantitative Invertibility and Approximation for the Truncated Hilbert and Riesz Transforms

In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase transformation, we prove the existence of convex integration solutions $u$ with higher Sobolev regularity, i.e., there exists $\theta _{0}>0$ such that $\nabla u \in W^{s,p}_{loc}( \mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2})$ for $s\in (0,1)$, $p\in (1,\infty )$ with $0< sp < \theta _{0}$. We also recall a construction which shows that in very specific situations with additional symmetry much better regularity properties hold.

https://link.springer.com/content/pdf/10.1007/s10659-018-09719-3.pdf

Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation

In this article, we continue our investigation of the variable
 coefficients thin obstacle problem which was initiated in
 [20], [21]. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove the higher order Holder regularity for the regular free boundary, if the associated coefficients are of the corresponding regularity. For the zero obstacle, this yields an improvement of a full derivative for the free boundary regularity compared to the
regularity of the coefficients. In the presence of inhomogeneities,
we gain three halves of a derivative for the free boundary
regularity with respect to the regularity of the inhomogeneity.
Further, we show analyticity of the regular free boundary for analytic
coefficients. We also discuss the set-up of $W^{1,p}$ coefficients
with $p>n+1$ and $L^p$ inhomogeneities. 
Key ingredients in our analysis are the introduction of generalized
Holder spaces, which allow to interpret the transformed fully
nonlinear, degenerate (sub)elliptic equation as a perturbation of the
Baouendi-Grushin operator, various uses of intrinsic geometries
associated with appropriate operators, the application of the
implicit function theorem to deduce (higher) regularity.

The Variable Coefficient Thin Obstacle Problem: Higher Regularity

In this paper we study a Landis-type conjecture for fractional Schrodinger equations of fractional power $s\in(0,1)$ with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for \emph{differentiable} potentials with some a priori bounds, if a solution decays at a rate $e^{-|x|^{1+}}$, then this solution is trivial. On the other hand, for $s\in(1/4,1)$ and merely bounded \emph{non-differentiable} potentials, if a solution decays at a rate $e^{-|x|^\alpha}$ with $\alpha>4s/(4s-1)$, then this solution must again be trivial. Remark that when $s\to 1$, $4s/(4s-1)\to 4/3$ which is the optimal exponent for the standard Laplacian. For the case of non-differential potentials and $s\in(1/4,1)$, we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.

Angkana Rüland

Papers

On Single Measurement Stability for the Fractional Calder\'on Problem.

Quantitative Invertibility and Approximation for the Truncated Hilbert and Riesz Transforms

Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation

The Variable Coefficient Thin Obstacle Problem: Higher Regularity

On the Fractional Landis Conjecture