Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent

Given a smooth bounded domain [Formula: see text] in [Formula: see text] with [Formula: see text], we study the existence and the profile of positive solutions for the following elliptic Neumann problem: [Formula: see text] where [Formula: see text] is a large exponent and [Formula: see text] denotes the outer unit normal vector to the boundary [Formula: see text]. For suitable domains [Formula: see text], by a constructive way we prove that, for any non-negative integers [Formula: see text], [Formula: see text] with [Formula: see text], if [Formula: see text] is large enough, such a problem has a family of positive solutions with [Formula: see text] boundary layers and [Formula: see text] interior layers which concentrate along [Formula: see text] distinct [Formula: see text]-dimensional minimal submanifolds of [Formula: see text], or collapse to the same [Formula: see text]-dimensional minimal submanifold of [Formula: see text] as [Formula: see text].

On the Discretization of the Power-Law Hemolysis Model

Flow-induced hemolysis remains a concern for blood-contacting devices, and computer-based prediction of hemolysis could facilitate faster and more economical refinement of such devices. While evaluation of convergence of velocity fields obtained by computational fluid dynamics (CFD) simulations has become conventional, convergence of hemolysis calculations is also essential. In this paper, convergence of the power-law hemolysis model is compared for simple flows, including pathlines with exponentially increasing and decreasing stress, in gradually expanding and contracting Couette flows, in a sudden radial expansion and in the Food and Drug Administration (FDA) channel. In the exponential cases, convergence along a pathline required from one to tens of thousands of timesteps, depending on the exponent. Greater timesteps were required for rapidly increasing (large exponent) stress and for rapidly decreasing (small exponent) stress. Example pathlines in the Couette flows could be fit with exponential curves, and convergence behavior followed the trends identified from the exponential cases. More complex flows, such as in the radial expansion and the FDA channel, increase the likelihood of encountering problematic pathlines. For the exponential cases, comparison of converged hemolysis values with analytical solutions demonstrated that the error of the converged solution may exceed 10% for both rapidly decreasing and rapidly increasing stress.

Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Let Ω ⊂ ℝ2 be a bounded domain with smooth boundary and b(x) > 0 a smooth function defined on ∂Ω. We study the following Robin boundary value problem: $$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$ where ν denotes the exterior unit vector normal to ∂Ω, 0 < λ < +∞ and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p → +∞ and simultaneously as λ → +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.