In this article, we prove the existence of weak solutions for the following nonlinear problem that contains a nonlocal operator −D s • (AD s (.)) which is defined by the distributional Riesz fractional derivatives and a measurable, bounded, positive definite matrix A(x).where Ω ⊂ R n is a bounded open set with a Lipschitz boundary, s ∈ (0, 1) and n > 2s. Under some suitable conditions on the nonlinear term f and the matrix A(x), it has shown that this problem has at least one weak solution u. We use in our proof the Leray-Schauder degree method to prove the existence of weak solutions and the Banach fixed point theorem to prove the uniqueness of weak solution in a particular case.
In this article, we study the existence of a weak solutions for the nonlinear fractional elliptic systems with Dirichlet boundary conditions in three cases. We use the Leray-Schauder degree and some sufficient conditions for the solvability of a resonance and non-resonance systems with respect to the spectrum of the fractional Laplacian.
Many authors used different techniques for image restoration. Variational approaches are mainly used for this purpose (see [4] and [5]). For example, the Mumfort-Shah is a model type for the free discontinuity in the restoration and segmentation of images. The main charateristic of this model is the isotropic diffusion, minimizing the length of the boundaries, while the classical models due to Rudin-Osher-Fatmi, minimize the total variation of the longest curves in the images (see [12] and references therein). In this work we propose a new partial differential equation of order four that can be used for image restoration. This proposed model is a combination of the fast growth with respect to low curvature and the slow growth when the gradient is large. Numerical results are presented for both second and fourth order partial differential equations.Keywords---Ill-posed problem, inverse problems, image de-noising, edge detection, nonlinear partial differential equations.
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