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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 | 27 | en |
dc.date.accessioned | 2007-01-22T13:53:20Z | - |
dc.date.available | 2007-01-22T13:53:20Z | - |
dc.date.issued | 2004 | - |
dc.identifier.citation | Winter, M. and Wei, J. (2004) 'Higher-Order Energy Expansions and Spike Locations', Calculus of Variations and Partial Differential Equations, 20(4), pp. 403-430. doi:10.1007/s00526-003-0240-y. | en |
dc.identifier.issn | The original publication is available at www.springerlink.com | - |
dc.identifier.uri | http://bura.brunel.ac.uk/handle/2438/558 | - |
dc.description.abstract | We consider the following singularly perturbed semilinear elliptic problem: (I)\left\{ \begin{array}{l} \epsilon^{2} \Delta u - u + f(u)=0 \ \ \mbox{in} \ \Omega, \\ u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \end{array} \right. where \Om is a bounded domain in R^N with smooth boundary \partial \Om, \ep>0 is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional J_\ep defined by J_\ep [u]:= \int_\Om \left(\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- F(u)\right) dx \ \ \ \ \ \mbox{for} \ u \in H^1 (\Om), where F(u)=\int_0^u f(s)ds. 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 at P_\ep \in \partial \Om. In this paper, we obtain a 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. Some applications of this expansion are given. | en |
dc.format.extent | 242415 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | en | - |
dc.publisher | Springer | en |
dc.subject | Higher-Order Energy Expansions, | en |
dc.subject | Ricci Curvature | en |
dc.title | Higher-Order Energy Expansions and Spike Locations | en |
dc.type | Research Paper | en |
dc.identifier.doi | https://doi.org/10.1007/s00526-003-0240-y | - |
Appears in Collections: | Dept of Mathematics Research Papers Mathematical Sciences |
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