We introduce variants of the variational image denoising method proposed by Blomgren et alinterpolates between totalvariation denoising and isotropic diffusion denoising. We study how parameter choices affect results and allow tuning between TV denoising and isotropic diffusion for respecting texture on one spatial scale while denoising features assumed to be noise on finer spatial scales. Furthermore, we prove existence and (where appropriate) uniqueness of minimizers. We consider both L 2 and L 1 data fidelity terms.

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A new method for solving boundary value problems for linear and integrable nonlinear PDEs has recently been introduced. For a linear evolution equation with dispersion relation ω(k), in a time-dependent concave domain, (t) < x < ∞, (t) < 0, this method yields the solution q(x, t) as a line integral in the complex k-plane, whose integrand has explicit (x, t)-dependence of the formFor a nonlinear integrable evolution equation the situation is similar, but the integrand also involves the solution M(x, t, k) to a RiemannHilbert problem whose jump function also has explicit (x, t)-dependence of exponential form. These representations for linear and for nonlinear evolution equations are convenient for the study of the long-time asymptotics, using the stationary phase and the Deift-Zhou methods, respectively. Here we study the long-time asymptotics in a time-dependent concave domain for a linear evolution equation with a spatial derivative of arbitrary order, and for the defocusing nonlinear Schrödinger equation. For completeness we also study the long-time asymptotics for a linear evolution equation with a spatial derivative of arbitrary order in the fixed domain 0 < x < ∞.

In this paper, we prove the existence and uniqueness of the solutions of the one-dimensional steady-state reduced semiconductor equations for the voltagedriven case using a two-piece doping function that jumps from −α to β.

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