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

Higher-Order Energy Expansions and Spike Locations

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

Clustered spots in the FitzHugh-Nagumo system

We construct {\bf clustered} spots for the following FitzHugh-Nagumo system: \left\{\begin{array}{l}\ep^2\Delta u +f(u)-\delta v =0\quad \mbox{in} \ \Om,\\[2mm]\Delta v+ u=0 \quad \mbox{in} \ \Om,\\[2mm] u= v =0 \quad\mbox{on} \ \partial \Om, \end{array} \right. where \Om is a smooth and bounded domain in $R^2$. More precisely, we show that for any given integer $K$, there exists an \ep_{K}>0 such that for 0<\ep <\ep_K,\, \ep^{m^{'}} \leq \delta \leq \ep^m for some positive numbers $m^{'}, m$, there exists a solution (u_{\ep},v_{\ep}) to the FitzHugh-Nagumo system with the property that u_{\ep} has $K$ spikes Q_{1}^\ep, ..., Q_K^\ep and the following holds: (i) The center of the cluster \frac{1}{K} \sum_{i=1}^K Q_i^\ep approaches a hotspot point Q_0\in\Om. (ii) Set l^\ep=\min_{i \not = j} |Q_i^\ep -Q_j^\ep| =\frac{1}{\sqrt{a}} \log\left(\frac{1}{\delta \ep^2 }\right) \ep ( 1+o(1)). Then (\frac{1}{l^\ep} Q_1^\ep, ..., \frac{1}{l^\ep} Q_K^\ep) approaches an optimal configuration of the following problem: {\it $(*) \ \ \$ Given $K$ points $Q_1, ..., Q_K \in R^2$ with minimum distance $1$, find out the optimal configuration that minimizes the functional $\sum_{i \not = j} \log |Q_i-Q_j|$}

A local quantum version of the Kolmogorov theorem

Consider in $L^2 (\R^l)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon Q_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector \om, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and \ep\in\R. Then there exists \ep^\ast >0 with the property that if |\ep|<\ep^\ast there is a diophantine frequency \om(\ep) such that all eigenvalues E_n(\hbar,\ep) of H(\ep) near 0 are given by the quantization formula E_\alpha(\hbar,\ep)= {\cal E}(\hbar,\ep)+\la\om(\ep),\alpha\ra\hbar +|\om(\ep)|\hbar/2 + \ep O(\alpha\hbar)^2, where $\alpha$ is an $l$-multi-index.Comment: 18 page

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

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