In the case of $K\ne \overline{D\left(A\right)}$, we study Cauchy problems and periodic problems for nonlinear evolution equation $u\left(t\right)\in K$, $u^{\prime }\left(t\right)+Au\left(t\right)\ni f\left(t,u\left(t\right)\right)$, $0\le t\le T$, where $A$ is a maximal monotone operator on a Hilbert space $H$, $K$ is a closed, convex subset of $H$, $V$ is a subspace of $H$, and $f:\left[0,T\right]\times \left(K\cap V\right)\to H$ is of Carathéodory type.

We prove an existence result for solution to a class of nonlinear degenerate elliptic equation associated with a class of partial differential operators of the form $Lu\left(x\right)={\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)$, with $D_j=\partial /\partial x_j$, where $a_{ij}:\Omega \to ℝ$ are functions satisfying suitable hypotheses.

We consider a two-phase system mainly in three dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasistatic Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean-field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer and Otto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking into account the geometry of the spatial distribution.

We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.

We study the existence and multiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions.