Some local eigenvalue estimates involving curvatures

Fall, Mouhamed Moustapha
November 2009
Calculus of Variations & Partial Differential Equations;Nov2009, Vol. 36 Issue 3, p437
Academic Journal
We first establish a local Faber–Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively.


Related Articles

  • CONTINUOUS DEPENDENCE OF EIGENVALUES OF p-BIHARMONIC PROBLEMS ON p. BENEDIKT, JIŘÍ // Communications on Pure & Applied Analysis;May2013, Vol. 12 Issue 3, p1469 

    We are concerned with the Dirichlet and Neumann eigenvalue problem for the ordinary quasilinear fourth-order (p-biharmonic) equation (|u"|p-2 u")" = λ|u|p-2 in [0, 1], p > 1. It is known that the eigenvalues of the Dirichlet and Neumann p-biharmonic problem are positive and nonnegative,...

  • A NEUMANN BOUNDARY-VALUE PROBLEM ON AN UNBOUNDED INTERVAL. AMSTER, PABLO; DEBOLI, ALBERTO // Electronic Journal of Differential Equations;2008, Vol. 2008, Special section p1 

    We study a Neumann boundary-value problem on the half line for a second order equation, in which the nonlinearity depends on the (unknown) Dirichlet boundary data of the solution. An existence result is obtained by an adapted version of the method of upper and lower solutions, together with a...

  • Boundary value problem for the Helmholtz equation outside cuts on the plane with boundary conditions of the third kind. Krutitskii, P. // Differential Equations;Oct2007, Vol. 43 Issue 10, p1420 

    The article investigates boundary value problems for the Helmholtz equation outside cuts on the plane in which a boundary condition of the third kind is posed on both sides of the cuts. The Dirichlet and Neumann problems outside cuts were considered. It describes the propagation of acoustic...

  • AN ALTERNATING POTENTIAL-BASED APPROACH TO THE CAUCHY PROBLEM FOR THE LAPLACE EQUATION IN A PLANAR DOMAIN WITH A CUT. Chapko, R.; Johansson, B. T. // Computational Methods in Applied Mathematics;2008, Vol. 8 Issue 4, p315 

    We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and...

  • Basic boundary value problems in complex analysis. Begehr, H. // Journal of Applied Functional Analysis;Jan2007, Vol. 2 Issue 1, p57 

    In order to develop a theory of boundary value problems for complex partial differential equations of arbitrary order at first the three basic boundary value problems for analytic functions and more generally for the inhomogeneous Cauchy-Riemann equation are investigated. They are the Schwarz,...

  • A GLOBAL APPROACH TO GROUND STATE SOLUTIONS. KORMAN, PHILIP // Electronic Journal of Differential Equations;2008, Vol. 2008, Special section p1 

    We study radial solutions of semilinear Laplace equations. We try to understand all solutions of the problem, regardless of the boundary behavior. It turns out that one can study uniqueness or multiplicity properties of ground state solutions by considering curves of solutions of the...

  • The Calculation of Annular Duct geometries by prescribing a Velocity Distribution on the Pressure Surface and the Radius of the Suction Surface. Pavlika, Vasos // International MultiConference of Engineers & Computer Scientists;2007, p2375 

    In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The velocity distribution is defined...

  • On the eigenvalue of the laplacian in a domain perforated along the boundary. Gadyl'shin, R. R.; Koroleva, Yu. O.; Chechkin, G. A. // Doklady Mathematics;Jun2010, Vol. 81 Issue 3, p337 

    The article investigates the Laplacian operator's eigenvalue. It provides information on the applicability of the Laplacian's eigenvalue in solving the elliptic operators' homogenized problems and cites the methods of computing the solutions' convergence rate. It entails the variational...

  • Finite size effects in the anisotropic (λ/4!)(φ[sub 1][sup 4]+φ[sub 2][sup 4])[sub d] model. Fosco, C. D.; Svaiter, N. F. // Journal of Mathematical Physics;Nov2001, Vol. 42 Issue 11, p5185 

    We consider the (λ/4!)(φ[sub 1][sup 4]+φ[sub 2][sup 4]) model on a d-dimensional Euclidean space, where all but one of the coordinates are unbounded. Translation invariance along the bounded coordinate, z, which lies in the interval [0,L], is broken because of the boundary conditions...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics