Hadamard Type Asymptotics for Eigenvalues of the Neumann Problem for Elliptic Operators

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. We consider two applications: the Laplacian in both $C^{1,\alpha}$ and Lipschitz domains. For the $C^{1,\alpha}$ case, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result. For the Lipschitz case, the proximity of eigenvalues is estimated

A Fixed Point Theorem in Locally Convex Spaces

For a locally convex space \X with the topology given by a family \{\pa{\cdot}\}_{\alpha \in \I} of seminorms, we study the existence and uniqueness of fixed points for a mapping \K \colon \DK \rightarrow \DK defined on some set \DK \subset \X. We require that there exists a linear and positive operator \KK, acting on functions defined on the index set \I, such that for every u,v \in \DK \pa{\K(u) - \K(v)} \leq \KK \apa{u-v}(\alpha) , \quad \alpha \in \I. Under some additional assumptions, one of which is the existence of a fixed point for the operator \KK + \pa[\cdot]{\K(0)}, we prove that there exists a fixed point of \K. For a class of elements satisfying \KK^{n} \apa{u}(\alpha) \rightarrow 0 as $n\rightarrow \infty$, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points. We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudo-differential equations with nonlinear terms

Radiographic Bone Loss and Its Relation to Patient-Specific Risk Factors, LDL Cholesterol, and Vitamin D: A Cross-Sectional Study

The influence of patient-specific factors such as medical conditions, low-density lipoprotein cholesterol (LDL-C) or levels of 25-hydroxyvitamin D (25OHD) on periodontal diseases is frequently discussed in the literature. Therefore, the aim of this retrospective cross-sectional study was to evaluate potential associations between radiographic bone loss (RBL) and patient-specific risk factors, particularly LDL-C and 25OHD levels. Patients from a dental practice, who received full-mouth cone beam CTs (CBCTs) and blood-sampling in the course of implant treatment planning, were included in this study. RBL was determined at six sites per tooth from CBCT data. LDL-C and 25OHD levels were measured from venous blood samples. Other patient-specific risk factors were assessed based on anamnesis and dental charts. Statistical analysis was performed applying non-parametric procedures (Mann–Whitney U tests, error rates method). Data from 163 patients could be included in the analysis. RBL was significantly higher in male patients, older age groups, smokers, patients with high DMFT (decayed/missing/filled teeth) score, lower number of teeth, and high LDL-C levels (≥160 mg/dL). Furthermore, patients with high 25OHD levels (≥40 ng/mL) exhibited significantly less RBL. In summary, RBL was found to be associated with known patient-specific markers, particularly with age and high LDL-C levels

Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach

This work is devoted to the equation ˆ u(y) dS(y) = f(x), x ∈ S, (1) |x − y| N−1 S where S is the graph of a Lipschitz function ϕ on R N with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local L p-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to th

Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach

This work is devoted to the equation where S is the graph of a Lipschitz function φ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lp-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted Lp- and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces. In Paper 2, we present a fixed point theorem for a locally convex space , where the topology is given by a family of seminorms. We study the existence and uniqueness of fixed points for a mapping defined on a set . It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every ,   Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p( ; · ), we prove that there exists a fixed point of . For a class of elements satisfying Kn (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms. In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 < p < ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2. In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on RN, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question

Continuous Nowhere Differentiable Functions

In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by $$W(x) = \sum_{k=0}^{\infty} a^k\cos(b^k\pi x)\text{,}$$ where $a$ is a real number with $0 1 + 3\pi/2$. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass' nor Riemann's function was the first such construction. The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differentiable. Around 1860, the Swiss mathematician Charles Cell\'erier also discovered (independently) an example which unfortunately wasn't published until 1890 (posthumously). After the publication of the Weierstrass function, many other mathematicians made their own contributions. We take a closer look at many of these functions by giving a short historical perspective and proving some of their properties. We also consider the set of all continuous nowhere differentiable functions seen as a subset of the space of all real-valued continuous functions. Surprisingly enough, this set is even ``large'' (of the second category in the sense of Baire).Validerat; 20101217 (root

Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electronâ\u80\u93electron interactions in the potential