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DC Field | Value | Language |
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dc.contributor.author | Winter, M | - |
dc.contributor.author | Wei, J | - |
dc.coverage.spatial | 6 | en |
dc.date.accessioned | 2007-01-22T15:55:41Z | - |
dc.date.available | 2007-01-22T15:55:41Z | - |
dc.date.issued | 2003 | - |
dc.identifier.citation | Winter, M. and Wei, J. (2003) 'Higher order energy expansions for some singularly perturbed Neumann problems', Comptes Rendus Mathematique, 337(1), pp. 37-42. doi:10.1016/s1631-073x(03)00269-3. | en |
dc.identifier.uri | http://bura.brunel.ac.uk/handle/2438/568 | - |
dc.description.abstract | We consider the following singularly perturbed semilinear elliptic problem: \epsilon^{2} \Delta u - u + u^p=0 \ \ \mbox{in} \ \Omega, \quad u>0 \ \ \mbox{in} \ \ \Omega \quad \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, where \Om is a bounded smooth domain in R^N, \ep>0 is a small constant and p is a subcritical exponent. Let J_\ep [u]:= \int_\Om (\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- \frac{1}{p+1} u^{p+1}) dx be its energy functional, where u \in H^1 (\Om). Ni and Takagi proved that for a single boundary spike solution u_\ep, the following asymptotic expansion holds J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + o(\ep)\Bigg], where c_1>0 is a generic constant, P_\ep is the unique local maximum point of u_\ep and H(P_\ep) is the boundary mean curvature function. In this paper, we obtain the following higher order expansion of J_\ep [u_\ep]: J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + \ep^2 [c_2 (H(P_\ep))^2 + c_3 R (P_\ep)]+ o(\ep^2)\Bigg], where c_2, c_3 are generic constants and R(P_\ep) is the Ricci scalar curvature at P_\ep. In particular c_3 >0. Applications of this expansion will be given. | en |
dc.format.extent | 127239 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | en | - |
dc.publisher | Elsevier | en |
dc.subject | Higher order expansions; Ricci curvature | en |
dc.subject | Singularly perturbed problem | en |
dc.title | Higher order energy expansions for some singularly perturbed Neumann problems | en |
dc.type | Research Paper | en |
dc.identifier.doi | https://doi.org/10.1016/s1631-073x(03)00269-3 | - |
Appears in Collections: | Dept of Mathematics Research Papers Mathematical Sciences |
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