本项目主要研究在图像处理中有深刻应用的1-Laplace方程解的存在性以及特征值问题。与p-Laplace方程不同，1-Laplace方程的解一般属于BV空间，是不连续的。同时由于BV空间的对偶空间尚不是很清楚，这为利用变分方法研究解的存在性带来了很大的困难，特别是在证明相应泛函（PS）序列的收敛性上。本项目的研究主要有两方面：（1）高维空间中特征值问题。高维空间中计算BV函数的全变差非常困难，主要利用轴对称化方法简化问题，再计算相应泛函的强斜率寻找强特征值。（2）临界增长的1-Laplace方程解的存在性。通过分析泛函弱斜率与次微分的关系，采用新方法（例如定义（Concrete-PS）序列或者证明Radon空间中的Brezis-Lieb引理）证明（PS）序列的收敛性，利用不光滑泛函的临界点理论来研究方程解的存在性，进一步，采用轴对称化方法计算一些具体方程解的表达式。

This research project mainly study the existence of solutions to 1-Laplace equations and eigenvalue problems, which has widely use in image processing. Differ to p-Lalace equations, the solutions to 1-Laplace equations are belong to BV space, they can be discontinous. Since the dual space of BV space is not completely understood, it bring many difficulties to the research if we want to use the variational methods to study the existence of solutions, especially to prove the convergence of (PS) sequences. The contents of the research is bifurcated. (1) eigenvalue problems in high dimensional space. The computation of total variation of a BV function is very difficult, so the method of symmetrization is applied to simplify problems, and then the strong slope of corresponding functional was computed to obtain strong eigenvalues. (2) existence of solutions to 1-Laplace equations at critical growth. The relationship between weak slope and subdifferentials of the functional was studied and a new method (for example, define a (Concrete-PS) sequence or prove Brezis-Lieb lemma in Radon measure space) was used to prove the convergence of (PS) sequence, then the nonsmooth critical point theory was employed to obtain the existence of solutions to equations, moreover, the method of symmetrization was employed to compute the formula of solutions to some special problems.