The Czochralski method of the industrial production of a silicon single crystal consists of pulling up the single crystal from the silicon melt. The flow of the melt during the production is called the Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function), heat convection-conduction equations, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect.

This paper deals with the analysis of the system in the form used for numerical simulation. The weak formulation is derived and the existence of the weak solution to the stationary and the evolution problem is proved.

We consider the obstacle problem $$\{\begin{array}{l}\text{minimize}I\left(u\right)={\int }_{\Omega }G\left(\nabla u\right)dx\text{among functions}u:\Omega \to R\\ \text{such}\text{that}u{|}_{\partial \Omega }=0\text{and}u\ge \Phi \text{a}\text{.e}\text{.}\end{array}$$ for a given function and a bounded Lipschitz domain $\Omega$ in ${\mathbf{R}}^n$. The growth properties of the convex integrand $G$ are described in terms of a $N$-function $A:\left[0,\infty \right)\to \left[0,\infty \right)$ with . If $n\le 3$, we prove, under certain assumptions on $G,C^{1,\infty }$-partial regularity for the solution to the above obstacle problem. For the special case where $A\left(t\right)=tln\left(1+t\right)$ we obtain $C^{1,\alpha }$-partial regularity when $n\le 4$. One of the main features of the paper is that we do not require any power growth of $G$.

We consider a certain class of quasilinear elliptic equations with a term in the critical growth range. We prove the existence of positive solutions in bounded and unbounded domains. The proofs involve several generalizations of standard variational arguments.

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

The existence of positive solutions to certain systems of ordinary differential equations is studied. Particular forms of these systems are satisfied by radial solutions of associated partial differential equations.

In this paper we introduce the uniform asymptotic normal structure and the uniform semi-Opial properties of Banach spaces. This part is devoted to a study of the spaces with these properties. We also compare them with those spaces which have uniform normal structure and with spaces with $WCS\left(X\right)>1$.

We consider a thermo-elastic plate system where the elastic equation does not account for rotational forces. We select the case of hinged mechanical B.C. and Neumann thermal B.C., which are coupled on the boundary. We show that the corresponding s.c. contraction semigroup (on a natural energy space) is analytic and, hence, uniformly stable. Because of the boundary (high) coupling, this case of B.C. is not contained in, and is more challenging than, recent known cases of the literature [L-R.1], [L-L.1], [L-T.1].

Under suitable conditions we are able to solve the semilinear wave equation in any dimension. We are also able to compute the essential spectrum of the linear wave operator for the rotationally invariant periodic case.

We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications to second-order Hamiltonian systems are given.

In this paper we will investigate the existence of multiple solutions for the problem $$\left(P\right)-{\Delta }_{p}u+g\left(x,u\right)={\lambda }_{1}h\left(x\right){\left|u\right|}^{p-2}u,\text{in}\Omega ,u\in H_0^{1,p}\left(\Omega \right)$$ where ${\Delta }_{p}u=\text{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)$ is the $p$-Laplacian operator, $\Omega \subseteq {ℝ}^N$ is a bounded domain with smooth boundary, $h$ and $g$ are bounded functions, $N\ge 1$ and . Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).

We study nonlinear semigroups of holomorphic mappings on certain domains in complex Banach spaces. We examine, in particular, their differentiability and their representations by exponential and other product formulas. In addition, we also construct holomorphic retractions onto the stationary point sets of such semigroups.

We obtain a parabolic analogue of the well-known maximum principle established by R. Finn for solutions of the minimal surface equation.