We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as epsilon -> 0 of the solutions u(epsilon) of the nonlinear equation div a(epsilon)(x, del u(epsilon)) = div b(epsilon), where both a(epsilon) and b(epsilon) oscillate rapidly on several microscopic scales and a(epsilon) satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W-0(1,p)(Omega), where 1 < p < infinity. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe's method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

This PhD thesis is focussed on some problems of great interest in applied mathematics. More precisely, we investigate some new questions in homogenization theory, which have been motivated by some concrete problems in tribology. From the mathematical point of view these questions are euqipped with scales of Reynolds equations with rapidly oscillating coefficients. In particular, in this PhD thesis we derive the corresponding homogenized (averaged) equations. We consider the Reynolds equations in both the stationary and unstationary forms to analyze the effect of surface roughness on the hydrodynamic performance of bearings when a lubricant is flowing through it. In addition we have successfully developed a reiterated homogenization (with three scales) procedure which makes it possible to efficiently study problems connected to hydrodynamic lubrication including shape, texture and roughness. Furthermore, we solve a linear parabolic initial-boundary value problem with singular coefficients in non-cylindrical domains. We accomplish this feat by developing a variant of Rothe's method to prove the existence and uniqueness of a weak solution to the parabolic problem. By combining the Rothe's method and the technique of two scale convergence we derive a homogenized equation for a linear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution. In Chapter 1 we describe some possible types of surfaces a bearing can take. Out of these we select two types and derive the appropriate Reynolds equations needed for their analysis. Chapter 2 is devoted to the derivation of the homogenized equations associated with the stationary forms of the compressible and incompressible Reynolds equations. We derive these homogenized equations by using the multiple scales expansion technique. In Chapter 3 the homogenized equations for the unstationary forms of the Reynolds equations are considered and some numerical results based on the homogenized equations are presented. In Chapter 4 we consider the equivalent minimization problem (varia- tional principle) for the unstationary Reynolds equation and use it to derive a homogenized minimization problem. Moreover, we obtain both the lower and upper bounds for the derived homogenized problem. Chapter 5 is devoted to studying the combined effect that arises due to shape, texture and surface roughness in hydrodynamic lubrication. This is accomplished by first studying a general class of problems that includes the incompressible Reynolds problem in both cartesian and cylindrical coordi- nate forms. In Chapter 6 we prove a homogenization result for the nonlinear equation $\mathrm{div}\,a(x,x/\varpeilson,x/\varepsilon^2,\nabla u_{\varepsilon})=\mathrm{div}\,b(x,x/\varpeilson,x/\varepsilon^2)$, where the coefficients are assumed to be periodic and a is monotone and continuous. This kind of problem has applications in hydrodynamic lubrication of surfaces with roughness on different length scales. In Chapter 7 a variant of Rothe's method is developed, discussed and used to prove existence and uniqueness result for linear parabolic problem with singular coefficients in non-cylindrical domains. In Chapter 8 we combine the Rothe method with a homogenization technique (two-scale convergence) to handle a general time-dependent lin- ear parabolic problem. In particular we prove that both the approximating sequence and the final approximate solution are unique. Finally, we derive a concrete homogenization algorithm on how to compute this homogenized solution.

Some new multidimensional Hardy-type inequalities involving arithmetic mean operators with general positive kernels are derived. Our approach is mainly to use a convexity argument and the results obtained improve some known results in the literature and, in particular, some recent results in [S. Kaijser, L. Nikolova, L.-E. Persson, A. Wedestig, Hardy-type inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403-417] are generalized and complemented.

Some new refined multidimensional Hardy-type inequalities for p 2 and their duals are derived and discussed. Moreover, these inequalities hold in the reversed direction when 1 < p 2. The results obtained are based mainly on some new results for superquadratic and subquadratic functions. In particular, our results further extend the recent results in [J.A. Oguntuase and L.-E. Persson, Refinement of Hardy's inequalities via superquadratic and subquadratic functions, J

This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both artesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution.Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

This paper is devoted to the effects of surface roughness in hydrodynamic lubrication. The numerical analysis of such problems requires a very fine mesh to resolve the surface roughness, hence it is often necessary to do some type of averaging. Previously, homogenization (a rigorous form of averaging) has been successfully applied to Reynolds type differential equations. More recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. In these pioneering works, it has been assumed that only one surface is rough. In this paper we develop these results to include the unstationary case where both surfaces may be rough. More precisely, we first use multiple-scale expansion to obtain a homogenization result for a class of variational problems including the variational formulation associated with the unstationary Reynolds equation. Thereafter, we derive lower and upper bounds corresponding to the homogenized (averaged) variational problem. The bounds reduce the numerical analysis, in that one only needs to solve two smooth problems, i.e. no local scale has to be considered. Finally, we present several examples, where it is shown that the bounds can be used to estimate the effects of surface roughness with very high accuracy.

This Licentiate thesis is focussed on some new questions in homogenization theory, which have been motivated by some concrete problems in tribology. From the mathematical point of view, these questions are equipped with scales of Reynolds equations with rapidly oscillating coefficients. In particular, in this Licentiate thesis we derive the corresponding homogenized (averaged) equation. We consider the Reynolds equations in both the stationary and unstationary forms to analyze the effect of surface roughness on the hydrodynamic performance of bearings when a lubricant is flowing through it. In Chapter 1 we describe the possible types of surfaces a bearing can take. Out of these, we select two types and derive the appropriate Reynolds equations needed for their analysis. Chapter 2 is devoted to the derivation of the homogenized equations, associated with the stationary forms of the compressible and incompressible Reynolds equations. We derive these homogenized equations by using the multiple scales expansion technique. In Chapter 3 the homogenized equations for the unstationary forms of the Reynolds equations are considered and some numerical results based on the homogenized equations are presented. In chapter 4 we consider the equivalent minimization problem for the unstationary Reynolds equation and use it to derive a homogenized minimization problem. Finally, we obtain both the lower and upper bounds for the derived homogenized problem.