1,393,580 research outputs found

### Higher order energy expansions for some singularly perturbed Neumann problems

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

### Erythropoietin production by fetal mouse liver cells in response to hypoxia and adenylate cyclase stimulation

This study was done to investigate aspects of control of extrarenal erythropoietin (Ep) production. To this end we studied the effects of three stimuli of renal Ep production in the adult, i.e. hypoxia, cobalt, and activation of adenylate cyclase on Ep generation by cultured fetal mouse liver cells. The fetal liver was taken as a model for extrarenal Ep production because this organ is considered the predominant site of extrarenal Ep production. We found that Ep production by the cells increased as the oxygen concentration was decreased in the incubation atmosphere from 20% to 1%. Cobalt (10(-4)-10(-5) M) had no effect on Ep production. Activation of adenylate cyclase by forskolin (10(-5) M) or isoproterenol (10(-5) M) greatly enhanced Ep production. These findings indicate that the Ep-stimulating effect of cobalt is specific for the kidney. However, oxygen depletion and activation of adenylate cyclase seem to be more general stimuli in Ep-producing cells. Furthermore we found that Ep production in hypoxia correlated with lactate formation in the cultured liver cells. This finding suggests that Ep production in fetal livers under hypoxic conditions parallels the shift from aerobic to anaerobic cellular energy metabolism

### First analytic correction beyond PFA for the electromagnetic field in sphere-plane geometry

We consider the vacuum energy for a configuration of a sphere in front of a plane, both obeying conductor boundary condition, at small separation. For the separation becoming small we derive the first next-to-leading order of the asymptotic expansion in the separation-to-radius ratio \ep. This correction is of order \ep. In opposite to the scalar cases it contains also contributions proportional to logarithms in first and second order, \ep \ln \ep and \ep (\ln \ep)^2. We compare this result with the available findings of numerical and experimental approaches.Comment: 20 pages, 1 figur
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