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European Mathematical Society Publishing House
2024-03-29 08:44:10
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=30&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)
30
2011
3
Symmetries of the Generalized Variational Functional of Herglotz for Several Independent Variables
Bogdana
Georgieva
Sofia University "St. Kliment Ohridski", SOFIA, BULGARIA
Variational symmeties, Herglotz variational principle, invariant functional, Herglotz
This paper provides a method for calculating the symmetry groups of the functional defined by the generalized variational principle of Herglotz in the case of several independent variables. Examples of calculating variational symmetry groups are given, including those for the non-conservative nonlinear Klein-Gordon equation, and for the equations describing the propagation of electromagnetic fields in a conductive medium.
Calculus of variations and optimal control; optimization
General
253
268
10.4171/ZAA/1434
http://www.ems-ph.org/doi/10.4171/ZAA/1434
On Compactness of Minimizing Sequences Subject to a Linear Differential Constraint
Stefan
Krömer
Universität Köln, KÖLN, GERMANY
$\mathcal{A}$-free integral functionals, weak-strong convergence, differential constraints
For $\Omega\subset \mathbb R^N$ open, we consider integral functionals of the form \begin{align*} \textstyle{F(u):=\int_\Omega f(x,u)\,dx}, \end{align*} defined on the subspace of $L^p$ consisting of those vector fields $u$ which satisfy the system $\mathcal{A} u=0$ on $\Omega$ in the sense of distributions. Here, $\mathcal{A}$ may be any linear differential operator of first order with constant coefficients satisfying Murat's condition of constant rank. The main results provide sharp conditions for the compactness of minimizing sequences with respect to the strong topology in $L^p$. Although our results hold for bounded domains as well, our main focus is on domains with infinite measure, especially exterior domains.
Calculus of variations and optimal control; optimization
Partial differential equations
General
269
303
10.4171/ZAA/1435
http://www.ems-ph.org/doi/10.4171/ZAA/1435
Multiplicity Results for Classes of Infinite Positone Problems
Eunkyung
Ko
Mississippi State University, MISSISSIPPI STATE, UNITED STATES
Eun Kyoung
Lee
Pusan National University, BUSAN, SOUTH KOREA
R.
Shivaji
Mississippi State University, MISSISSIPPI STATE, UNITED STATES
Singular boundary value problems, infinite positone problems, multiplicity of positive solutions, sub-supersolutions
We study positive solutions to the singular boundary value problem \begin{equation*}\left\{\begin{alignedat}{2}-\Delta_p u &= \lambda \frac{f(u)}{u^\beta}& \quad &\mbox{in}~\Omega \\ u &= 0 & \quad &\mbox{on}~\partial \Omega, \end{alignedat}\right. \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $ p > 1, \lambda > 0, \beta \in (0,1)$ and $ \Omega$ is a bounded domain in $\mathbb{R}^{N}, N \geq 1.$ Here $f:~[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0.$ We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $\alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.
Partial differential equations
General
305
318
10.4171/ZAA/1436
http://www.ems-ph.org/doi/10.4171/ZAA/1436
A Note on Homogenization of Advection-Diffusion Problems with Large Expected Drift
Patrick
Henning
Universität Münster, MÜNSTER, GERMANY
Mario
Ohlberger
Universität Münster, MÜNSTER, GERMANY
Advection-diffusion, large drift, homogenization, two-scale convergence
This contribution is concerned homogenization of linear advection-diffusion problems with rapidly oscillating coefficient functions and large expected drift. Even though the homogenization of this type of problems is generally well known, there are several details that have not yet been treated explicitly or even not been treated at all. Here, we will have a special look at uniqueness, regularity, boundedness and equivalent formulations of the homogenized equation. In particular, we generalize results of Allaire and Raphael [C. R. Math. Acad. Sci. Paris 344 (2007)(8), 523–528] and Donato and Piatnitski [Multi Scale Problems and Asymptotic Analysis. Tokyo: Gakkotosho 2006, pp. 153–165]. The results obtained in this contribution are of special interest for the numerical analysis of multi-scale schemes to approximate the analytic solutions.
Partial differential equations
General
319
339
10.4171/ZAA/1437
http://www.ems-ph.org/doi/10.4171/ZAA/1437
An Approximation Result in Generalized Anisotropic Sobolev Spaces and Applications
Mostafa
Bendahmane
Universidad de Concepción, CONCEPCION, CHILE
Moussa
Chrif
Université Moulay Ismail, MEKNÈS, MOROCCO
Said
El Manouni
Al Imam University (IMSIU), RIYADH, SAUDI ARABIA
Approximation, anisotropic Sobolev space, segment property, strongly nonlinear elliptic problems
In this paper, we give an approximation result in some anisotropic Sobolev space. We also describe the action of some distributions in the dual and we mention two applications to some strongly nonlinear anisotropic elliptic boundary value problems.
Partial differential equations
General
341
353
10.4171/ZAA/1438
http://www.ems-ph.org/doi/10.4171/ZAA/1438
Concentration-Compactness Principle for Generalized Trudinger Inequalities
Robert
Černý
Charles University, PRAGUE 8, CZECH REPUBLIC
Petr
Gurka
Czech University of Life Sciences, PRAGUE 6, CZECH REPUBLIC
Stanislav
Hencl
Charles University, PRAGUE 8, CZECH REPUBLIC
Orlicz-Sobolev spaces, Concentration-Compactness
Let $\Omega\subset\mathbb R^n$, $n\geq 2$, be a bounded domain and let $\alpha < n-1$. We prove the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space $W^1_0L^n\log^{\alpha}L(\Omega)$ into the Orlicz space with the Young function $\exp\big(t^{\frac{n}{n-1-\alpha}}\big)-1$.
Functional analysis
Calculus of variations and optimal control; optimization
General
355
375
10.4171/ZAA/1439
http://www.ems-ph.org/doi/10.4171/ZAA/1439