- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 09:40:51
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=28&iss=3&update_since=2024-03-29
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
28
2009
3
Generalized Rademacher–Stepanov Type Theorem and Applications
Alireza
Ranjbar-Motlagh
Sharif University of Technology, TEHRAN, IRAN
Rademacher and Stepanov theorems, Sobolev and bounded variation spaces, generalized differentiability, Lipschitz manifolds, Orlicz spaces
The main purpose of this article is to generalize a theorem of Stepanov which provides a necessary and sufficient condition for almost everywhere differentiability of functions over Euclidean spaces. We state and prove an Lp -type generalization of the Stepanov theorem and then we extend it to the context of Orlicz spaces. Then, this generalized Rademacher–Stepanov type theorem is applied to the Sobolev and bounded variation maps with values into a metric space. It is shown that several generalized differentiability type theorems are valid for the Sobolev maps from a Lipschitz manifold into a metric space. As a byproduct, it is shown that the Sobolev spaces of Korevaar–Schoen and Reshetnyak are equivalent.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
249
275
10.4171/ZAA/1384
http://www.ems-ph.org/doi/10.4171/ZAA/1384
On Nontrivial Solutions of Variational-Hemivariational Inequalities with Slowly Growing Principal Parts
Vy Khoi
Le
Missouri University of Science and Technology, ROLLA, UNITED STATES
Dumitru
Motreanu
Université de Perpignan, PERPIGNAN, FRANCE
Variational-hemivariational inequality, Orlicz–Sobolev space, Mountain Pass theorem, linking theorem
This paper is concerned with the inclusion −div(a(|∇u|)∇u) + ∂u G(x, u) ∋ 0 in Ω, with Dirichlet boundary condition u = 0 on ∂Ω, in the case where the higher order part has slow growth and the lower order part is locally Lipschitz. By using a Mountain Pass theorem for variational-hemivariational inequalities without the Palais–Smale condition in Orlicz–Sobolev spaces, we show the existence of nontrivial solutions of the above inclusion.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
277
293
10.4171/ZAA/1385
http://www.ems-ph.org/doi/10.4171/ZAA/1385
Asymptotic Behavior of Approximate Solutions to Evolution Equations in Banach Spaces
Sergiu
Aizicovici
Ohio University, ATHENS, UNITED STATES
Simeon
Reich
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Alexander
Zaslavski
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Complete uniform space, convex function, descent method, generic property, initial value problem
We study evolution equations in Banach spaces governed by a class of mappings associated with continuous descent methods for the minimization of convex functions. In our previous work we showed that for most of these mappings (in the sense of Baire category) the corresponding solutions converged. In the present paper we show that this remains true even for approximate solutions.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
295
303
10.4171/ZAA/1386
http://www.ems-ph.org/doi/10.4171/ZAA/1386
On the Mathematical Analysis and Numerical Approximation of a System of Nonlinear Parabolic PDEs
J.
Kačur
Comenius University, BRATISLAVA, SLOVAK REPUBLIC
B.
Malengier
Universiteit Gent, GENT, BELGIUM
R.
Van Keer
Universiteit Gent, GENT, BELGIUM
Relaxation method, method of characteristics, contaminant transport, convection-diffusion with adsorption
In this paper we consider a boundary value problem for a system of 2 nonlinear parabolic PDEs e.g. arising in the context of flow and transport in porous media. The flow model is based on tho nonlinear Richard’s equation problem and is combined with the transport equation through saturation and Darcy’s velocity (discharge) terms. The convective terms are approximated by means of the method of characteristics initiated by P. Pironneau [Num. Math. 38 (1982), 871–885] and R. Douglas and T. F. Russel [SIAM J. Num. Anal. 19 (1982), 309–332]. The nonlinear terms in Richard’s equation are approximated by means of a relaxation scheme applied by W. Jäger and J. Kačur [RAIRO Model. Math. Anal. Num. 29 (1995), 605–627] and J. Kačur [IMA J. Num. Anal. 19 (1999), 119–154; SIAM J. Num. Anal. 39 (1999), c 290–316]. The convergence of the approximation method is proved.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
305
332
10.4171/ZAA/1387
http://www.ems-ph.org/doi/10.4171/ZAA/1387
Local Tb Theorem on Spaces of Homogeneous Type
Chaoqiang
Tan
Shantou University, SHANTOU, GUANGDONG, CHINA
Lixin
Yan
Zhongshan University, GUANGZHOU, GUANGDONG, CHINA
Local Tb theorem, singular integral operators, trees, BCR algorithm, spaces of homogeneous type
In this article, we obtain a local Tb theorem for singular integral operators on spaces of homogeneous type by using tree selection algorithm of the dyadic model and the BCR algorithm, which extends an earlier result of M. Christ [Colloq. Math. 60/61 (1990), 601–628].
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
333
347
10.4171/ZAA/1388
http://www.ems-ph.org/doi/10.4171/ZAA/1388
Existence Theory for Steady Flows of Fluids with Pressure and Shear Rate Dependent Viscosity, for Low Values of the Power-Law Index
Miroslav
Bulíček
Charles University, PRAGUE 8, CZECH REPUBLIC
V.
Fišerová
Technische Hochschule Darmstadt, DARMSTADT, GERMANY
Existence, weak solution, incompressible fluid, Lipschitz approximation of Sobolev functions, pressure- and shear-dependent material coefficients, decomposition of the pressure
We deal with a system of partial differential equations describing a steady flow of a homogeneous incompressible non-Newtonian fluid with pressure and shear rate dependent viscosity subject to the homogeneous Dirichlet (no-slip) boundary condition. We establish a global existence of a weak solution for a certain class of such fluids in which the dependence of the viscosity on the shear rate is polynomiallike, characterized by the power-law index. A decomposition of the pressure and Lipschitz approximations of Sobolev functions are considered in order to obtain almost everywhere convergence of the pressure and the symmetric part of the velocity gradient and thus obtain new existence results for low value of the power-law index.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
349
371
10.4171/ZAA/1389
http://www.ems-ph.org/doi/10.4171/ZAA/1389