Laplace Operator (laplace + operator)

Distribution by Scientific Domains


Selected Abstracts


Conformally invariant powers of the Dirac operator in Clifford analysis

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010
David Eelbode
Abstract The paper deals with conformally invariant higher-order operators acting on spinor-valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these are unique, up to a constant multiple. A general construction for such an invariant operators on manifolds with a given conformal spin structure was described in (Conformally Invariant Powers of the Ambient Dirac Operator. ArXiv math.DG/0112033, preprint), generalizing the case of powers of the Laplace operator from (J. London Math. Soc. 1992; 46:557,565). Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator (see Laplacian Operators and Curvature on Conformally Einstein Manifolds. ArXiv: math/0506037, 2006). Here we shall treat the spinor case on the sphere. We shall compute the explicit form of such operators on the sphere, and we shall show that they coincide with operators studied in (J. Four. Anal. Appl. 2002; 8(6):535,563). The methods used are coming from representation theory combined with traditional Clifford analysis techniques. Copyright © 2010 John Wiley & Sons, Ltd. [source]


Minimal regularity of the solutions of some transmission problems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2003
D. Mercier
We consider some transmission problems for the Laplace operator in two-dimensional domains. Our goal is to give minimal regularity of the solutions, better than H1, with or without conditions on the (positive) material constants. Under a monotonicity or quasi-monotonicity condition on the constants (or on the inverses according to the boundary conditions), we study the behaviour of the solution near vertex and near interior nodes and show in each case that the given regularity is sharp. Without condition we prove that the regularity near a corner is of the form H1+,, where , is a given bound depending on the material constants. Numerical examples are presented which confirm the sharpness of our lower bounds. Copyright © 2003 John Wiley & Sons, Ltd. [source]


A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface

MATHEMATISCHE NACHRICHTEN, Issue 9 2010
Giuseppe Cardone
Abstract It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the essential spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


On the number of interior peak solutions for a singularly perturbed Neumann problem

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2007
Fang-Hua Lin
We consider the following singularly perturbed Neumann problem: where , = , ,2/,x is the Laplace operator, , > 0 is a constant, , is a bounded, smooth domain in ,N with its unit outward normal ,, and f is superlinear and subcritical. A typical f is f(u) = up where 1 < p < +, when N = 2 and 1 < p < (N + 2)/(N , 2) when N , 3. We show that there exists an ,0 > 0 such that for 0 < , < ,0 and for each integer K bounded by where ,N, ,, f is a constant depending on N, ,, and f only, there exists a solution with K interior peaks. (An explicit formula for ,N, ,, f is also given.) As a consequence, we obtain that for , sufficiently small, there exists at least [,N, ,f/,N (|ln ,|)N] number of solutions. Moreover, for each m , (0, N) there exist solutions with energies in the order of ,N,m. © 2006 Wiley Periodicals, Inc. [source]


Regularity of the obstacle problem for a fractional power of the laplace operator

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2007
Luis Silvestre
Given a function , and s , (0, 1), we will study the solutions of the following obstacle problem: u , , in ,n, (,,)su , 0 in ,n, (,,)su(x) = 0 for those x such that u(x) > ,(x), lim|x| , + ,u(x) = 0. We show that when , is C1, s or smoother, the solution u is in the space C1, , for every , < s. In the case where the contact set {u = ,} is convex, we prove the optimal regularity result u , C1, s. When , is only C1, , for a , < s, we prove that our solution u is C1, , for every , < ,. © 2006 Wiley Periodicals, Inc. [source]