Partial Differential Equations Commons^™

Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in R^N (N > 1), centered at the origin and λ > 0. Here ƒ: R→R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ(t)/(tƒ'(t)−ƒ(t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ(t) = tp − k, k > 0 and 1 < p < (N + 2)/(N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].

Wing-Section Optimization For Supersonic Viscous Flow, Cem Cihan Item

Mechanical & Aerospace Engineering Theses & Dissertations

The recent interest in the High Speed Civil Transport (HSCT) has resulted in renewed research studies of optimized supersonic cruise transport configurations. Incorporation of flow viscosity effects in the design process of such a supersonic wing is currently under investigation. This may lead to more accurate problem formulations and, in tum, greater aerodynamic efficiency than can be obtained by the traditional, inviscid, linear theories. In this context, for a design code to be a candidate for a complex optimization problem, such as three-dimensional viscous supersonic wing design, it should be validated using simpler building-block shapes.

To optimize the shape of …

A Convergent Reconstruction Method For An Elliptic Operator In Potential Form, Lester Caudill

Department of Math & Statistics Faculty Publications

We investigate the problem of recovering a potential q(x) in the equation -∆u + q(x)u = 0 from overspecified boundary data on the unit square in R². The potential is characterized as a fixed point of a nonlinear operator, which is shown to be a contraction on a ball in C^∝. Uniqueness of q(x) follows, as does convergence of the resulting recovery scheme. Numerical examples, demonstrating the performance of the algorithm, are presented.