| 1. |
- Englis, Miroslav, et al.
(författare)
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On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions
- 2004
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Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; 570, s. 89-129
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Tidskriftsartikel (refereegranskat)abstract
- In an earlier work, J.-L. Lions produced a means for creating reproducing formulas for the space of harmonic functions with Sobolev boundary values on a domain in real Euclidean space. The present authors give a new proof of this result and also generalize the Lions formula to handle spaces of functions that are annihilated by an elliptic operator. The method of constructing the reproducing kernels seems to be based on the old paradigm of Aronszajn and Bergman. It is interesting to note that this model---that the kernel should take the form $$ K(x,y)=\sum_j e_j(x)·e_j(y) $$ for a suitable orthonormal basis $\{e_j\}$---goes back to the thesis of Bochner. That thesis well predates the early work of Bergman and Szegö.
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| 2. |
- Lukkassen, Dag, et al.
(författare)
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On some iterated means arising in homogenization theory
- 2004
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 49:4, s. 343-356
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Tidskriftsartikel (refereegranskat)abstract
- We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.
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| 3. |
- Almqvist, Andreas, et al.
(författare)
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New concepts of homogenization applied in rough surface hydrodynamic lubrication
- 2007
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Ingår i: International Journal of Engineering Science. - : Elsevier BV. - 0020-7225 .- 1879-2197. ; 45:1, s. 139-154
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Tidskriftsartikel (refereegranskat)abstract
- This work introduces a new concept of homogenization that enables efficient analysis of the effects of surface roughness representations obtained by measurements in applications modeled by the Reynolds equation. Examples of such applications are trust- and journal-bearings. The numerical analysis of these types of applications requires an extremely dense computational mesh in order to resolve the surface roughness, suggesting some type of averaging. One such method is homogenization, which has been applied to Reynolds type equations with success recently. This approach is similar to the technique proposed by Patir and Cheng, who introduced flow factors determining the hydrodynamic action due to surface roughness. The difference is, however, that the present technique has a rigorous mathematical support. Moreover, the recipe to compute the averaged coefficients is simple without any ambiguities. Using either the technique proposed by Patir and Cheng or homogenization, the coefficients determining the averaged Reynolds equation are obtained by solving differential equations on a local scale. Unfortunately, this is detrimental when investigating the effects induced by real, measured, surface roughness, even though these local problems may be solved in parallel. The present work presents a solution by applying the technique based on bounds. This technique transforms the stationary Reynolds equation into two computationally feasible forms, one for the upper bound and one for the lower bound, where the flow factors are obtained by straightforward integration. Together with the preciseness of these bounds, the bounds approach becomes an eminent tool suitable for investigating the effect of real, measured, surface roughness on hydrodynamic performance.
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| 4. |
- Høibakk, Ralph, et al.
(författare)
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On some power means and their geometric constructions
- 2018
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Ingår i: Mathematica Æterna. - : Hilaris Ltd. - 1314-3336 .- 1314-3344. ; 8:3, s. 139-158
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Tidskriftsartikel (refereegranskat)abstract
- The main aim of this paper is to further develop the recently initiatedresearch concerning geometric construction of some power means wherethe variables are appearing as line segments. It will be demonstratedthat the arithmetic mean, the harmonic mean and the quadratic meancan be constructed for any number of variables and that all power meanswhere the number of variables are n = 2m, m 1 2 N for all powersk = 2?q and k = 2q; q 2 N can be geometrically constructed.
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| 5. |
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| 6. |
- Lions, Jacque-Louis, et al.
(författare)
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Reiterated homogenization of monotone operators
- 2000
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Ingår i: Comptes rendus de l'Académie des sciences. Série 1, Mathématique. - 0764-4442 .- 1778-3577. ; 330:8, s. 675-680
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Tidskriftsartikel (refereegranskat)abstract
- In this Note we study reiterated homogenization of nonlinear equations of the form −div(a(x/,x/2,Du))=f, where a is periodic in the first two arguments and monotone in the third. We state that u converges weakly in W1,p(Ω) (and even in some multiscale sense), as →0, to the solution u0 of a limit problem. Moreover, we give an explicit expression for the limit problem.
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| 7. |
- Lions, Jacques-Louis, et al.
(författare)
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Reiterated homogenization of nonlinear monotone operators
- 2001
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Ingår i: Chinese Annals of Mathematics. Series B. - 0252-9599 .- 1860-6261. ; 22B:1, s. 1-12
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, the authors study reiterated homogenization of nonlinear equations of the form -div(a(x, x/ε, x/ε2,Duε)) = f, where a is periodic in the first two arguments and monotone in the third. It is proved that uε converges weakly in W1,p(Ω) (and even in some multiscale sense), as ε → 0 to the solution u0 of a limit problem. Moreover, an explicit expression for the limit problem is given. The main results were also stated in [15]. This article presents the complete proofs of these results
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| 8. |
- Lukkassen, Dag, et al.
(författare)
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A new approach for estimating the friction in thin film lubrication
- 2010
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Ingår i: Mathematical problems in engineering (Print). - : Hindawi Limited. - 1024-123X .- 1563-5147. ; 2010
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Tidskriftsartikel (refereegranskat)abstract
- An important problem in the theory of lubrication is to model and analyze the effect of surface roughness on, for example, the friction and load carrying capacity. A direct numerical computation is often impossible since an extremely fine mesh is required to resolve the surface roughness. This suggests that one applies some averaging technique. The branch in mathematics which deals with this type of questions is known as homogenization. In this paper we present a completely new method for computing the friction. The main idea is that we study the variational problem corresponding to the Reynolds equation. We prove that the homogenized variational problem is closely related to the homogenized friction. Finally we use bounds on the homogenized Lagrangian to derive bounds for the friction. That these bounds can be used to efficiently compute the friction is demonstrated in a typical example
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| 9. |
- Lukkassen, Dag, et al.
(författare)
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Analysis of the effects of rough surfaces in compressible thin film flow by homogenization
- 2011
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Ingår i: International Journal of Engineering Science. - : Elsevier BV. - 0020-7225 .- 1879-2197. ; 49:5, s. 369-377
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Tidskriftsartikel (refereegranskat)abstract
- A classical problem in lubrication theory is to predict the pressure distribution in a thin fluid film between two surfaces which are in relative motion. If one of the surfaces is rough, then the distance between the surfaces is rapidly oscillating. This leads to that the governing Reynolds partial differential equation involves rapidly oscillating coefficients. The branch in mathematics which considers such types of equations is known as homogenization. In this paper we study the effects of surface roughness for a special type of compressible fluid. In particular, we derive homogenization results connected to the friction force and the load carrying capacity
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| 10. |
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