We consider singular perturbation variational problemsdepending on a small parameter ε. The right hand side is suchthat the energydoes not remain bounded as ε → 0. The asymptoticbehavior involves internallayers where most of the energy concentrates. Three examples are addressed,with limits elliptic, parabolic and hyperbolic respectively, whereas theproblems with ε > 0 are elliptic. In the parabolic and hyperboliccases, thepropagation of singularities appear as an integral property after integratingacross the layers.

Let W be a non-negative function of class C3 from $\xR^2$ to $\xR$, which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy $$ E_1(v)=\int_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x $$ is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and z-, it is possible to prove that there exists a function u from $\xR^2$ to itself which satisfies the equation $$ -\Delta u + \xDif W(u)^\mathsf{T}=0, $$ and the boundary conditions $$ \lim_{x_2\to +\infty} u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf{a}} \lim_{x_2\to -\infty} u(x_1,x_2)=z_-(x_1-m_-), \lim_{x_1\to -\infty}u(x_1,x_2)=\mathbf{a},\phantom{z_+(x_1-m_+)} \lim_{x_1\to+\infty}u(x_1,x_2)=\mathbf{b}. $$ The above convergences are exponentially fast; the numbers m+ and m- are unknowns of the problem.

We present a technique for the rapid and reliable prediction oflinear-functionaloutputs of elliptic coercive partial differential equations with affineparameter dependence. The essential components are (i )(provably) rapidlyconvergent global reduced-basis approximations – Galerkin projectiononto a spaceW N spanned by solutions of the governing partial differentialequation at Nselected points in parameter space; (ii ) a posteriori error estimation– relaxations of the error-residual equation that provideinexpensive bounds for the error in the outputs of interest; and (iii ) off-line/on-line computational procedures – methods whichdecouple the generationand projection stages of the approximation process. The operationcount for theon-line stage – in which, given a new parameter value, we calculatethe output ofinterest and associated error bound – depends only on N (typicallyvery small) andthe parametric complexity of the problem; the method is thus ideallysuited for therepeated and rapid evaluations required in the context of parameter estimation,design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basisapproximations of elliptic coercive equations. The resulting errorestimates are, in some cases, quite sharp: the ratio of the estimatederror in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, inother cases, the necessary “bound conditioners” – in essence,operator preconditioners that (i ) satisfy an additional spectral“bound” requirement, and (ii ) admit the reduced-basisoff-line/on-line computational stratagem – either can not be found, oryield unacceptably large effectivities. In this paper we introduce a newclass of improved bound conditioners: the critical innovation is thedirect approximation of the parametric dependence of theinverse of the operator (rather than the operator itself); wethereby accommodate higher-order (e.g., piecewise linear) effectivityconstructions while simultaneously preserving on-line efficiency. Simpleconvex analysis and elementary approximation theory suffice to prove thenecessary bounding and convergence properties.

In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.