Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian

In the paper, we consider the following hemivariational inequality problem involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda u\in \alpha (x) \partial F(x,u) & x \in \varOmega , \\ u=0 & x\in \mathbb{R} ^{N} \backslash \varOmega , \end{cases} $${(−Δ)su+λu∈α(x)∂F(x,u)x∈Ω,u=0x∈RN∖Ω, where Ω is a bounded smooth domain in $\mathbb{R} ^{N}$RN with $N\geq 3$N≥3, $(-\Delta )^{s}$(−Δ)s is the fractional Laplacian with $s\in (0,1)$s∈(0,1), $\lambda >0$λ>0 is a parameter, $\alpha (x): \varOmega \rightarrow \mathbb{R} $α(x):Ω→R is a measurable function, $F(x, u):\varOmega \times \mathbb{R} \rightarrow \mathbb{R} $F(x,u):Ω×R→R is a nonsmooth potential, and $\partial F(x,u)$∂F(x,u) is the generalized gradient of $F(x, \cdot )$F(x,⋅) at $u\in \mathbb{R} $u∈R. Under some appropriate assumptions, we obtain the existence of a nontrivial solution of this hemivariational inequality problem. Moreover, when F is autonomous, we obtain the existence of infinitely many solutions of this problem when the nonsmooth potentials F have suitable oscillating behavior in any neighborhood of the origin (respectively the infinity) and discuss the properties of the solutions.

A Improvement for GNC Method of Nonconvex Nonsmooth Image Restoration

Nonconvex nonsmooth potential functions have superior restoration performance for the images with neat boundaries. But the nondifferentiality could cause many numerical difficulties. Thus the graduated nonconvex (GNC) method is suggested to deal with these problems. In this paper, a class of nonconvex nonsmooth approximate potential functions have been constructed, which can help our get a better initial value of the original problem. The numerical results show the restored perfprmance of the proposed methods.