An Extension Problem Related to the Fractional Laplacian

Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)] characterized the fractional Laplacian (−Δ) s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of L p norm using the Caffarelli–Silvestre’s extension technique.

Well-posedness of a fractional porous medium equation on an evolving surface

The algebraic and algorithmic study of integro-differential algebras and operators has only started in the past decade. Integro-differential operators allow us in particular to study initial value and boundary problems for linear ODEs from an algebraic point of view. Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods. Adding integral operators and evaluations, many new phenomena appear, including zero devisors and non-finitely generated ideals. In this tutorial, we give an introduction to symbolic methods for integro-differential operators and boundary problems developed over the last years. In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients. We will also outline methods for manipulating and solving linear boundary problems and illustrate them with an implementation.

The present work is devoted to calculating a first regularized trace of one integro-differential operator with the main part of the Sturm-Liouville type on a segment with punctured points at integral perturbation of “transmission” conditions. The integro-differential Sturm-Liouville operator −y″(x)+q(x)y(x)+γ∫0πy(t)dt=λy(x) given on the segments πn(k−1)<x<πnk, k=1,n¯; n ≥ 2 is considered. Boundary conditions of the Dirichlet type: y(0) = 0, y(π) = 0 are given on the left-hand and right-hand ends of the segment [0, π]. The functions continuous on [0, π], the first derivatives of which have jumps at the points x=πnk, are solutions. The value of jumps is expressed by the formula y′(πkn−0)=y′(πkn+0)−βk∫0πy(t)dt, k=1,n−1¯. The basic result of the paper is the exact formula of the first regularized trace of the considered differential operator.

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an extension problem in which the boundary is of codimension two.

The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.

The paper is devoted to calculation of a first regularized trace of one integro-differential operator with the main part of the Sturm-Liouville type on a segment with punctured points at integral perturbation of “transmission” conditions. The Sturm-Liouville operator −y″(x)+q(x)y(x)+γ∫0πy(t)dt=λy(x) given on the segments πn(k−1)<x<πnk,k=1,n¯;n≥2 is considered. Boundary conditions of the Dirichlet type: y(0) = 0, y(π) = 0 are given on the left-hand and right-hand ends of the segment [0, π]. The functions are continuous on [0, π], the first derivatives of which have jumps at the points x=πnk are solutions. The value of jumps is expressed by the formula y′(πkn−0)=y′(πkn+0)−βk∫0πy(t)dt, k=1, n−1¯. The basic result of the paper is the exact formula of the first regularized trace of the considered differential operator.

The investigation of the properties of the integro-differential operators will be carried out. Which generalizes the well-known Bavrin operators to the fractional value of the parameters. The properties of the defined operators are in the classes of the polyharmonic operators. It is established that the newly defined fractional operators map the polyharmonic functions on the ball to the polyharmonic functions. Also it is proposed that the inverse for the fractional operator and application of the integro-differential fractional operators to solve biharmonic problems with fractional boundary conditions. The sufficient condition for existence and uniqueness of the solution for biharmonic equation with fractional boundary conditions are obtained. The solution of the biharmonic equation is obtained by using the integro-differential fractional operator.

We considered the issues arising when Daugman’s integrodifferential operator is used to localize iris edges. Daugman’s integro-differential operator is a widely common method to localize the iris, however, problems of the operator optimization poorly presented in the literature. The article considers existing methods of optimization of the operator applied at the iris recognition step. Based on the provided research we proposed using Nelder-Mead and Differential Evolution methods to optimize the integro-differential operator. The problems of the iris localization and their solving methods were considered. The article focuses attention on non-cooperative iris recognition. The very general boundary conditions based on iris’s anatomy which are not dependent on captured image properties were defined. The results of the comparative analysis of the accuracy and performance of selected optimization methods of Daugman’s integrodifferential operator were presented at the experimental results chapter. It was found that the Differential evolution optimization method gives fine performance and correctness. It was concluded that the Differential evolution is expedient as Daugman’s integro-differential operator optimization method.

We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian. We show that the quantum behavior of such a nonlocal field theory in $d$-dimensions can be described in terms of a local action in $d+1$ dimensions which can be quantized using the canonical operator formalism though giving up local commutativity. In particular, we discuss how to obtain the two-point correlation functions and the vacuum energy density of the nonlocal fractional theory as a brane limit of the bulk correlators. We show explicitly how the quantized extension problem reproduces exactly the same particle content of other approaches based on the spectral representation of the fractional propagator. We also briefly discuss the inverse fractional Laplacian and possible applications of this approach in general relativity and cosmology.

We establish the local well-posedness of the Bartnik static metric extension problem for arbitrary Bartnik data that perturb that of any sphere in a Schwarzschild $$\{t=0\}$$ { t = 0 } slice. Our result in particular includes spheres with arbitrary small mean curvature. We introduce a new framework to this extension problem by formulating the governing equations in a geodesic gauge, which reduce to a coupled system of elliptic and transport equations. Since standard function spaces for elliptic PDEs are unsuitable for transport equations, we use certain spaces of Bochner-measurable functions traditionally used to study evolution equations. In the process, we establish existence and uniqueness results for elliptic boundary value problems in such spaces in which the elliptic equations are treated as evolutionary equations, and solvability is demonstrated using rigorous energy estimates. The precise nature of the expected difficulty of solving the Bartnik extension problem when the mean curvature is very small is identified and suitably treated in our analysis.

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of one-dimensional regular Schrodinger operators is presented. It is shown that the set of all self-adjoint extensions of one-dimensional regular Schrodinger operators is in one-to-one correspondence with the group of unitary operators on the finite-dimensional Hilbert space of boundary data, and they are characterized by a generalized class of boundary conditions that include the well-known Dirichlet, Neumann, Robin, and (quasi-)periodic boundary conditions. The numerical algorithm is based on a nonlocal boundary extension of the finite element method and their convergence is proved. An appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions and the conditioning of its computation is analyzed. Some significant numerical experiments are also discussed as well as the comparison with some standard algorithms. In particular it is shown that appropriate perturbations of stand...

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

The fractional Laplacian operator $$(-\varDelta )^s$$(-Δ)s on a bounded domain $$\varOmega $$Ω can be realized as a Dirichlet-to-Neumann map for a degenerate elliptic equation posed in the semi-infinite cylinder $$\varOmega \times (0,\infty )$$Ω×(0,?). In fact, the Neumann trace on $$\varOmega $$Ω involves a Muckenhoupt weight that, according to the fractional exponent $$s$$s, either vanishes $$(s 1/2)$$(s>1/2). On the other hand, the normal trace of the solution has the reverse behavior, thus making the Neumann trace analytically well-defined. Nevertheless, the solution develops an increasingly sharp boundary layer in the vicinity of $$\varOmega $$Ω as $$s$$s decreases. In this work, we extend the technology of automatic $$hp$$hp-adaptivity, originally developed for standard elliptic equations, to the energy setting of a Sobolev space with a Muckenhoupt weight, in order to accommodate for the problem of interest. The numerical evidence confirms that the method maintain exponential convergence. Finally, we discuss image denoising via the fractional Laplacian. In the image processing community, the standard way to apply the fractional Laplacian to a corrupted image is as a filter in Fourier space. This construction is inherently affected by the Gibbs phenomenon, which prevents the direct application to spliced images. Since our numerical approximation relies instead on the extension problem, it allows for processing different portions of a noisy image independently and combine them, without complications induced by the Gibbs phenomenon.