Large exponent behavior for power-type nonlinear evolution equations and applications
- PDF / 511,724 Bytes
- 34 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 127 Views
Journal of Evolution Equations
Large exponent behavior for power-type nonlinear evolution equations and applications Qing Liu
Abstract. Motivated by math models of image denoising and collapsing sandpiles, we are concerned with asymptotic behavior for a class of fully nonlinear power-type evolution equations in one space dimension as the exponent tends to infinity. It turns out that an initial layer appears in the large exponent limit. In order to examine the initial layer, we rescale the solution by stretching the time variable and study a fully nonlinear equation with a discontinuous and unbounded parabolic operator. We establish uniqueness and existence of viscosity solutions to this limit equation.
1. Introduction This paper is a continuation of [24] on large exponent behavior for power curvature flow equation arising in image processing. The power mean curvature flow describes the motion of a surface in Rn governed by the law V = Hα , where α > 0 is a given component, V denotes the normal velocity and H stands for the mean curvature of the surface. Here, Hα should be understood as |H|α−1 H so as to maintain the parabolicity of the problem for any α > 0. We refer to [3,4,26,27] and many others for results on various geometric properties on this flow. In the level set formulation, the geometric motion is expressed as ∇u α−1 ∇u =0 div u t − |∇u| div |∇u| |∇u|
in Rn × (0, ∞)
(1.1)
with a Lipschitz initial value u(·, 0) = u 0 in Rn . It is well known that there exists a unique viscosity solution u α to this problem; for well-posedness results, we refer to [12] and [15] in the case α = 1 and to [18,20] for a general α > 0. Applications of this equation to image processing are elaborated in [1,2,10,25]. Motivated by applications in image denoising [10], we investigate in [24] the large exponent behavior for (1.1) under the assumption that the level sets of u 0 are all convex. Mathematics Subject Classification: 35K93, 53C44, 35D40, 35B40 Keywords: Viscosity solutions, Asymptotic behavior, Power curvature flow.
Q. Liu
J. Evol. Equ.
It is shown [24] that the solution converges locally uniformly in Rn × (0, ∞) to the minimal solution U of the obstacle problem ∇U + 1, U − u 0 = 0 in Rn min − div |∇U | as α → ∞. In the case when n = 2, this result can be interpreted in image processing as the effect of instantaneous noise removal. The large exponent behavior becomes much more complicated when the convexity assumption on u 0 is dropped. In the present work, we restrict to the case n = 2 and study the nonconvex case, assuming in addition that each moving level curve can be expressed as the graph of a function in R. In other words, we consider α−1 u uxx xx 2 21 = 0 in R × (0, ∞). (1.2) u t − (1 + u x ) 3 3 (1 + u 2 ) 2 (1 + u 2 ) 2 x
x
We aim to rigorously understand the limit behavior of this equation when α → ∞. We must point out that although (1.2) has a clear and natural geometric meaning, well posedness of the associated Cauchy problem for a Lipschitz initial valu
Data Loading...
