Asymptotic Analysis - Volume 22, issue 2

Authors: Bellettini, Giovanni | Fragalà, Ilaria

Article Type: Research Article

Abstract: We approximate a hypersurface \varSigma with prescribed anisotropic mean curvature with solutions u _{\varepsilon} of suitable nonlinear elliptic equations depending on a small parameter \varepsilon>0 . We work in relative geometry, by endowing \mathbb{R}^N with a Finsler norm \phi describing the anisotropy. The main result states that \varSigma and \{u _{\varepsilon}=0\} are close of order \varepsilon^2\vert\log\varepsilon\vert^2 , and this estimate is optimal. This is obtained for two different elliptic equations by sub‐ and supersolutions technique, under smoothness and nondegeneracy assumptions on \varSigma . Basic steps …are: (i) an explicit computation of the second variation of the \phi ‐Minkowski content along geodesics; (ii) the definition of a Laplace–Beltrami operator on \varSigma ; (iii) the expansion of the \phi ‐mean curvature of \varSigma in a suitable tubular neighbourhood. Show more

Keywords: Finsler spaces, surfaces with prescribed mean curvature, second variation, asymptotic behaviour of solutions of PDE’s, nonlinear elliptic equations

Citation: Asymptotic Analysis, vol. 22, no. 2, pp. 87-111, 2000

Authors: Skinner, L.A.

Article Type: Research Article

Abstract: A uniformly valid asymptotic expansion is established for the solution to certain singularly perturbed generalized Abel integral equations. The results are applied to a problem in heat conduction.

Keywords: Volterra integral equation, singular perturbation, uniformly valid expansion

Citation: Asymptotic Analysis, vol. 22, no. 2, pp. 113-127, 2000

Authors: Yakubov, Sasun

Article Type: Research Article

Abstract: We investigate the Regge problem in Sobolev spaces with a weight. Moreover, conditions of proved theorems are weaker than those in the author’s paper published in the Journal of Mathematical Analysis and Applications 220 (1998), pp. 224–249. We prove an isomorphism, coerciveness with a defect with respect to the spectral parameter and 2‐fold completeness of root functions of the Regge problem. By the suggested method one can investigate both irregular and regular problems.

Keywords: Regge problem, coerciveness, 2‐fold completeness, root functions, boundary‐functional conditions

Citation: Asymptotic Analysis, vol. 22, no. 2, pp. 129-148, 2000

Authors: Pistoia, Angela

Article Type: Research Article

Abstract: We prove the existence of symmetric solutions to the singularly perturbed semilinear Dirichlet or Neumann problem \cases{-\varepsilon^2\Delta w+w=f(w) \quad \mbox{in}\ \varOmega,\cr w>0\quad \mbox{in}\ \varOmega,\cr \curpartial w/\curpartial\nu=0\quad \mbox{or}\quad w=0\quad \mbox{on}\ \curpartial\varOmega} under certain assumptions on the nonlinearity f .

Keywords: Semilinear Neumann problem, semilinear Dirichlet problem, singular perturbation, symmetry of solutions

Citation: Asymptotic Analysis, vol. 22, no. 2, pp. 149-161, 2000

Authors: Li, Feng‐Quan

Article Type: Research Article

Abstract: In this paper, in the case that the equivalued surface boundary \tilde{\varGamma}_1^\varepsilon shrinks to a fixed point on boundary \varGamma_1 , we discuss the limit behaviour of solutions for a class of boundary value problems with equivalued surface for hyperbolic equations.

Keywords: hyperbolic equations, boundary value problem with equivalued surface, limit behaviour

Citation: Asymptotic Analysis, vol. 22, no. 2, pp. 163-176, 2000