相关论文: The Laplacian on the unit square in a self-similar…

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate…

We study the eigenvalues of the Laplacian on ellipsoids that are obtained as perturbations of the standard Euclidean unit sphere in dimension two. A comparison of these eigenvalues with those of the standard Euclidean unit sphere is…

In this paper, we establish an analogue of the classical mean value property for both the harmonic functions and some general functions in the domain of the Laplacian on the Sierpinski gasket. Furthermore, we extend the result to some other…

We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside.…

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with…

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate…

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the…

We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and the 3-dimensional Sierpinski gasket, but the method is expected to be effective for…

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpi\'nski carpet and the higher dimensional Menger sponges, but also a…

We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the…

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving…

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…

We consider Laplacian Growth of self-similar domains in different geometries. Self-similarity determines the analytic structure of the Schwarz function of the moving boundary. The knowledge of this analytic structure allows us to derive the…

We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedrons: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices,…

We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The…

In this paper, we compute universal estimates of eigenvalues of a coupled system of elliptic differential equations in divergence form on a bounded domain in Euclidean space. As an application, we show an interesting case of rigidity…

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram…

The Laplacian plays a major role in the mathematical analysis of partial differential equations. Recently, the work of J. Kigami, taken up by R. S. Strichartz, allowed the construction of an operator of the same nature, defined locally, on…

Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…