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Mateusz Kwaśnicki - One of the best experts on this subject based on the ideXlab platform.
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extension technique for complete bernstein functions of the Laplace Operator
Journal of Evolution Equations, 2018Co-Authors: Mateusz Kwaśnicki, Jacek MuchaAbstract:We discuss the representation of certain functions of the Laplace Operator \(\Delta \) as Dirichlet-to-Neumann maps for appropriate elliptic Operators in half-space. A classical result identifies \((-\Delta )^{1/2}\), the square root of the d-dimensional Laplace Operator, with the Dirichlet-to-Neumann map for the \((d + 1)\)-dimensional Laplace Operator \(\Delta _{t,x}\) in \((0, \infty ) \times \mathbf {R}^d\). Caffarelli and Silvestre extended this to fractional powers \((-\Delta )^{\alpha /2}\), which correspond to Operators \(\nabla _{t,x} (t^{1 - \alpha } \nabla _{t,x})\). We provide an analogous result for all complete Bernstein functions of \(-\Delta \) using Krein’s spectral theory of strings. Two sample applications are provided: a Courant–Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrodinger Operators \(\psi (-\Delta ) + V(x)\), as well as an upper bound for the eigenvalues of these Operators. Here \(\psi \) is a complete Bernstein function and V is a confining potential.
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extension technique for complete bernstein functions of the Laplace Operator
arXiv: Analysis of PDEs, 2017Co-Authors: Mateusz Kwaśnicki, Jacek MuchaAbstract:We discuss representation of certain functions of the Laplace Operator $\Delta$ as Dirichlet-to-Neumann maps for appropriate elliptic Operators in half-space. A classical result identifies $(-\Delta)^{1/2}$, the square root of the $d$-dimensional Laplace Operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace Operator $\Delta_{t,x}$ in $(0, \infty) \times \mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-\Delta)^{\alpha/2}$, which correspond to Operators $\nabla_{t,x} (t^{1 - \alpha} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-\Delta$ using Krein's spectral theory of strings. Two sample applications are provided: a Courant--Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schr\"odinger Operators $\psi(-\Delta) + V(x)$, as well as an upper bound for the eigenvalues of these Operators. Here $\psi$ is a complete Bernstein function and $V$ is a confining potential.
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fractional Laplace Operator and meijer g function
Constructive Approximation, 2017Co-Authors: Bartlomiej Dyda, Alexey Kuznetsov, Mateusz KwaśnickiAbstract:We significantly expand the number of functions whose image under the fractional Laplace Operator can be computed explicitly. In particular, we show that the fractional Laplace Operator maps Meijer G-functions of \(|x|^2\), or generalized hypergeometric functions of \(-|x|^2\), multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the Operator \((1-|x|^2)_+^{\alpha /2} (-\Delta )^{\alpha /2}\) with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace Operator in the unit ball in a companion paper (Dyda et al., Eigenvalues of the fractional Laplace Operator in the unit ball, 2015, arXiv:1509.08533).
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eigenvalues of the fractional Laplace Operator in the unit ball
Journal of The London Mathematical Society-second Series, 2017Co-Authors: Bartlomiej Dyda, Alexey Kuznetsov, Mateusz KwaśnickiAbstract:We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace Operator (−Δ)α/2 in the unit ball D⊂Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while the lower bounds involve the lesser known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace Operator applied to a linearly dense set of functions in L2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper (B. Dyda, A. Kuznetsov and M. Kwaśnicki, ‘Fractional Laplace Operator and Meijer G-function’, Constr. Approx., to appear, doi:10.1007/s00365-016-9336-4). Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d⩽2 and α∈(0,2], or d⩽9 and α=1, and we provide strong numerical evidence for d⩽9 and general α∈(0,2].
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fractional Laplace Operator and meijer g function
arXiv: Analysis of PDEs, 2015Co-Authors: Bartlomiej Dyda, Alexey Kuznetsov, Mateusz KwaśnickiAbstract:We significantly expand the number of functions whose image under the fractional Laplace Operator can be computed explicitly. In particular, we show that the fractional Laplace Operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the Operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace Operator in the unit ball in a companion paper "Eigenvalues of the fractional Laplace Operator in the unit ball".
T A Bolokhov - One of the best experts on this subject based on the ideXlab platform.
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Properties of the Radial Part of the Laplace Operator for l=1 in a Special Scalar Product
Journal of Mathematical Sciences, 2016Co-Authors: T A BolokhovAbstract:We develop self-adjoint extensions of the radial part of the Laplace Operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ^3 to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse Operator of the radial part in question. Bibliography: 8 titles.
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Extensions of the Quadratic Form of the Transverse Laplace Operator
Journal of Mathematical Sciences, 2016Co-Authors: T A BolokhovAbstract:We study the quadratic form of the Laplace Operator in 3 dimensions written in spherical coordinates and acting on transverse components of vector-functions. Operators which act on parametrizing functions of one of the transverse components with angular momentum 1 and 2 appear to be fourth-order symmetric Operators with deficiency indices (1, 1). We consider self-adjoint extensions of these Operators and propose the corresponding extensions for the initial quadratic form. The relevant scalar product for angular momentum 2 differs from the original product in the space of vector-functions, but, nevertheless, it is still local in radial variable. Eigenfunctions of the Operator extensions in question can be treated as stable soliton-like solutions of the corresponding dynamical system whose quadratic form is a functional of the potential energy.
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properties of the l 1 radial part of the Laplace Operator in a special scalar product
arXiv: Spectral Theory, 2015Co-Authors: T A BolokhovAbstract:We develop self-adjoint extensions of the l=1 radial part of the Laplace Operator in a special scalar product. The product arises as the transfer of the plain product from R^3 into the set of functions parametrizing one of the two components of the transverse vector field. The similar extensions are treated for the square of inverse Operator of the radial part in question.
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extensions of the quadratic form of the transverse Laplace Operator
arXiv: Spectral Theory, 2014Co-Authors: T A BolokhovAbstract:We review the quadratic form of the Laplace Operator in 3 dimensions in spehrical coordinates which acts on the transverse components of vector functions. Operators, acting on the parametrizing functions of one of the transverse components with angular momentum 1 and 2, appear to be fourth order symmetric differential Operators with deficiency indices (1,1). We develop self-adjoint extensions of these Operators and propose correspondent extensions for the initial quadratic form. The relevant scalar product for the angular momentum 2 differs from the original product in the space of vector functions, but nevertheless it is still local in radial variable.
T A Shaposhnikova - One of the best experts on this subject based on the ideXlab platform.
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homogenization of variational inequality for the Laplace Operator with nonlinear constraint on the flow in a domain perforated by arbitrary shaped sets critical case
Journal of Mathematical Sciences, 2018Co-Authors: T A Shaposhnikova, M N ZubovaAbstract:We construct and justify a homogenized model of the variational inequality with the Laplace Operator and a nonlinear boundary constraint on the flow on arbitrary shaped cavities generating perforation of the domain with critical values of parameters.
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homogenization of a variational inequality for the Laplace Operator with nonlinear restriction for the flux on the interior boundary of a perforated domain
Nonlinear Analysis-real World Applications, 2014Co-Authors: Willi Jager, Maria Neussradu, T A ShaposhnikovaAbstract:Abstract In this paper, we study the asymptotic behavior of solutions u e of the elliptic variational inequality for the Laplace Operator in domains periodically perforated by balls with radius of size C 0 e α , C 0 > 0 , α ∈ ( 1 , n n − 2 ] , and distributed with period e . On the boundary of the balls, we have the following nonlinear restrictions u e ≥ 0 , ∂ ν u e ≥ − e − γ σ ( x , u e ) , u e ( ∂ ν u e + e − γ σ ( x , u e ) ) = 0 , γ = α ( n − 1 ) − n . The weak convergence of the solutions u e to the solution of an effective problem is given. In the critical case α = n n − 2 , the effective equation contains a nonlinear term which has to be determined as a solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is proved.
E Elizalde - One of the best experts on this subject based on the ideXlab platform.
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functional determinant of the massive Laplace Operator and the multiplicative anomaly
Journal of Physics A, 2015Co-Authors: Guido Cognola, E Elizalde, Sergio ZerbiniAbstract:After a brief survey of zeta function regularization issues and of the related multiplicative anomaly, illustrated with a couple of basic examples, namely the harmonic oscillator and quantum field theory at finite temperature, an application of these methods to the computation of functional determinants corresponding to massive Laplacians on spheres in arbitrary dimensions is presented. Explicit formulas are provided for the Laplace Operator on spheres in dimensions and for 'vector' and 'tensor' Laplacians on the unitary sphere S4.
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zeta function for the Laplace Operator acting on forms in a ball with gauge boundary conditions
Communications in Mathematical Physics, 1997Co-Authors: E Elizalde, M Lygren, D V VassilevichAbstract:The Laplace Operator acting on antisymmetric tensor fields in a D-dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the Operator are found explicitly for all values of D. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour and pole compensation, the zeta function of the Operator is obtained. From its expression, in particular, ζ(0) and ζ'(0) are evaluated exactly. A table is given in the paper for D=3,4,...,8. The functional determinants and Casimir energies are obtained for D=3,4,...,6.
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zeta function for the Laplace Operator acting on forms in a ball with gauge boundary conditions
arXiv: High Energy Physics - Theory, 1996Co-Authors: E Elizalde, M Lygren, D V VassilevichAbstract:The Laplace Operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the Operator are found explicitly for all values of $D$. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour, and pole compensation, the zeta function of the Operator is obtained. From its expression, in particular, $\zeta (0)$ and $\zeta'(0)$ are evaluated exactly. A table is given in the paper for $D=3, 4, ...,8$. The functional determinants and Casimir energies are obtained for $D=3, 4, ...,6$.
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heat kernel coefficients of the Laplace Operator on the d dimensional ball
Journal of Mathematical Physics, 1996Co-Authors: M Bordag, E Elizalde, Klaus KirstenAbstract:We present a very quick and powerful method for the calculation of heat kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non‐trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane. We apply our method to the case of the heat kernel expansion of the Laplace Operator on a D‐dimensional ball with either Dirichlet, Neumann or, in general, Robin boundary conditions. The final formulas are quite simple. Using this case as an example, we illustrate in detail our scheme —which serves for the calculation of an (in principle) arbitrary number of heat kernel coefficients in any situation when the basis functions are known. We provide a complete list of new results for the coefficients B3,..., B10, corresponding to the D‐dimensional ball with all the mentioned boundary conditions and D=3,4,5.
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zeta function determinant of the Laplace Operator on the d dimensional ball
arXiv: High Energy Physics - Theory, 1995Co-Authors: M Bordag, B Geyer, Klaus Kirsten, E ElizaldeAbstract:We present a direct approach for the calculation of functional determinants of the Laplace Operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, $D$, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions $D=2,3,4,5$ and $6$.
D V Vassilevich - One of the best experts on this subject based on the ideXlab platform.
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zeta function for the Laplace Operator acting on forms in a ball with gauge boundary conditions
Communications in Mathematical Physics, 1997Co-Authors: E Elizalde, M Lygren, D V VassilevichAbstract:The Laplace Operator acting on antisymmetric tensor fields in a D-dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the Operator are found explicitly for all values of D. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour and pole compensation, the zeta function of the Operator is obtained. From its expression, in particular, ζ(0) and ζ'(0) are evaluated exactly. A table is given in the paper for D=3,4,...,8. The functional determinants and Casimir energies are obtained for D=3,4,...,6.
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zeta function for the Laplace Operator acting on forms in a ball with gauge boundary conditions
arXiv: High Energy Physics - Theory, 1996Co-Authors: E Elizalde, M Lygren, D V VassilevichAbstract:The Laplace Operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the Operator are found explicitly for all values of $D$. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour, and pole compensation, the zeta function of the Operator is obtained. From its expression, in particular, $\zeta (0)$ and $\zeta'(0)$ are evaluated exactly. A table is given in the paper for $D=3, 4, ...,8$. The functional determinants and Casimir energies are obtained for $D=3, 4, ...,6$.
