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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
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
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