Research On Theoretical Problems Of Partial Differential Equations In The Background Of Image Restoration

Image restoration is an important part of digital image processing,which using the available surrounding information to fill in the damaged portions.It is often used to repair small areas,text deletions and hiding some objects.PDE-based image restoration techniques are a class of new image processing techniques,rapidly developed in image processing and analysis area in recent years.Because it can overcome some difficult problems that are difficult to handle by the classical image restoration method,so that they have become an active research topics in image processing.Its basic idea is to transform the image processing problem into the minimization of the energy functional,by looking for the minimization point of energy functional,to obtain the solution of the original image processing problem.In this paper,we investigate several kinds of minimization problems of variational functionals,through using functional analysis theory,the basic theory of partial differential equations,and nonlinear analysis theory and methods,the existence and multiple solutions of several kinds of variable exponential partial differential equations and fractional partial differential equations are studied.The mean results are as follows:(1)Fourth order nonlinear eigenvalue problems involving variable exponents.We have obtained similar results under the following two cases:1)indefinite potential;2)Zero potential.Firstly,using Weierstrass theorem and some inequality techniques,established the existence of two positive constants ?0 and ?1 with ?0 ??1;Secondly,the spectral structure of the variable exponent eigenvalue problem is portrayed,that is,any ??[?,+?)is an eigenvalue,while and ??(0,?0)is not an eigenvalue.At the same time,an interesting question concerns the existence of eigenvalue in the interval[?0,?1)when ?0??1?(2)Elliptic partial differential equation involving the p(x)-Laplacian in RN.It is well known,the main difficulty in treating elliptic problem in RN with variable exponent arises from the lack of compactness.To overcome the difficulty,many scholars have devoted themselve to new technique,for example,Radial symmetry Technology[95,96],potential function[97-99],etc.In this chapter,does not need coerciveness hypothesis and not necessarily radially symmetric on the potential V.We established the existence of infinitely nontrivial solution when F(x,u)satisfies sublinear growth at infinity.(3)Superlinear problems involving p(x)-Laplacian-like operators.In this part,we consider the case that the nonlinearity is superlinear but does not satisfy the Ambrosett-Rabinowitz condition,the corresponding Euler-Lagrange variational function is not bound from below,many nonlinear analysis techniques can't be used.The main idea is to use the mountain-pass theorem under the(C)condition and to show that for any A>0,there is a sequence {?n}n=1 ?(?)RN and a sequence {un}n=1 ?(?)W0 1,p(x)(?)with ?n ? ?,c?n?c?,I?n(un)?c?I'?n(un)?0,and then prove that the u is a critical point of I? for critical value c?.According to this idea,for any ?>0,this problem has at least one nontrivial solution.(4)Kirchhoff type partial differential equations involving p(x)-Laplacian operators.The corresponding variational function is not bound from below,so it is natural to divide N(?)into three subset corresponding to local minimum,local maximum and points of inflection of fibering maps,variational function is bounded below on the corresponding submanifold,the main purpose of this paper is to study the existence of two positive solutions.(5)Nonlinear elliptic equation driven the fractional Laplacian.Using the Fourier transform,defining the Laplacian of fractional derivatives.Firstly,by applying a version of the three critical points theorem we obtain the existence of at least three solutions of the problem in H?(?).Secondly,by applying the Weierstrass Theorem and Mountain Pass Theorem,the existence of at least two nontrivial solutions are also been obtained when 2<?<2?*.