Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case (original) (raw)

Analytic and geometric aspects of Laplace operator on Riemannian manifold

Malaya Journal of Matematik

In the past decade there has been a flurry of work at intersection of spectral theory and Riemannian geometry. In this paper we present some of recent results on abstract spectral theory depending on Laplace-Beltrami operator on compact Riemannian manifold. Also, we will emphasize the interplay between spectrum of operator and geometry of manifolds by discussing two main problems (direct and inverse problems) with an eye towards recent developments.

On Transforming the Laplace Operator

In this communication it is shown that a function of the Laplace operator acting on an arbitrary function can be transformed to a three-dimensional integral. The cases of the exponential function and of an arbitrary function expressible as a power series, are treated. Two special cases of radial functions are presented based on elementary observations made here. 1

Convolution operators in the geometric function theory

Journal of Inequalities and Applications, 2012

The study of operators plays a vital role in mathematics. To define an operator using the convolution theory, and then study its properties, is one of the hot areas of current ongoing research in the geometric function theory and its related fields. In this survey-type article, we discuss historic development and exploit the strengths and properties of some differential and integral convolution operators introduced and studied in the geometric function theory. It is hoped that this article will be beneficial for the graduate students and researchers who intend to start work in this field. MSC: 30C45; 30C50

Laplace operators on the cone of Radon measures

Journal of Functional Analysis, 2015

We consider the infinite-dimensional Lie group G which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold X and the commutative multiplicative group of functions on X. The group G naturally acts on the space M(X) of Radon measures on X. We would like to define a Laplace operator associated with a natural representation of G in L 2 (M(X), µ). Here µ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure µ is not quasi-invariant with respect to the action of the group G. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group G (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on M(X) whose generator is the Laplace operator.

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

Results in Mathematics, 2011

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich-Lebedev and generalized Mehler-Fock transforms. Mapping properties and norm estimates in weighted Lp-spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L2.

N-dimensional Laplace transformations and their applications in partial differential equations

CHAPTER 3. FURTHER NEW RESULTS ON N-DEMENSIONAL LAPLACE AND INVERSE LAPLACE TRANSFORMATIONS 3.1. Introduction 3.2. The Image of Functions with the Argument ' 89 3.2.1. Applications of Theorem 3.2.1 3.2.2. Laplace Transforms of some Elementary and Special Functions with n Variables ^ 98 3.3. The Original of Functions with the Argument ^ 106 3.3.1. Examples Based Upon Theorem 3.3.1 Ill 3.4. The Image of Functions with the Argument 2pi(x 114 3.4.1 Applications of Theorem 3.4.1 119 3.5. The Original of Functions with the Argument 121 3.5.1. Example Based Upon Theorem 3.5.1 125 CHAPTER 4. THE SOLUTION OF INITIAL-BOUNDARY-VALUE PROBLEMS (IBVP'S) BY DOUBLE LAPLACE TRANSFORMATIONS 127 4.1. Introduction 127 4.2. Non-homogenous Linear Partial Differential Equations (PDEs) of the First Order 129 4.2.1. Partial Differential Equations of Type Ux+u.y = f{x,y),Q<x<«o,0<y<oo 129 4.2.2. Partial Differential Equations of Type au^ + buy + eeu = fix,y), 0<a:<<», 0<y<~ 136 4.3. Non-homogenous Second Order Linear Partial Differential V Equations of Hyperbolic Type 4.3.1. Partial Differential Equations of Type Ujcy = f(x,y),),0<x«>o,0<y<oo 4.3.2. The Wave Equation 4.4. Non-homogenous Second Order Partial Differential Equations of Parabolic Type 4.4.1. Partial Differential Equations of Type u"+2u^+Uyy + KU = fix,y\0<x<oo,0<y<oo CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS 154 5.1. Conclusions 154 5.2. Future Directions 155

Multi-dimensional Laplace transforms and systems of partial differential equations

International Mathematical Forum, 2006

The object of this paper is to establish new theorem and corollary involving systems of two-dimensional Laplace transforms containing several equations. This system can be used to calculate new Laplace transform pairs. In the second part, system of partial differential equations related to telegraph equation is solved by using the double Laplace transformation.