By Josef Bemelmans

An summary of the state-of-the-art in very important topics, together with - along with elliptic and parabolic matters - geometry, loose boundary difficulties, fluid mechanics, evolution difficulties usually, calculus of adaptations, homogenization, regulate, modelling and numerical research. The papers represent the lawsuits of the Fourth eu convention on Elliptic and Parabolic difficulties, held in Rolduc, the Netherlands, and Gaeta, Italy, in 2001.

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Additional info for Elliptic and Parabolic Problems: Rolduc and Gaeta 2001 : Proceedings of the 4th European Conference Rolduc, Netherlands 18-22 June 2001 : Gaeta, Italy 24-28 September 2001

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T e (0,T), the restriction of ft(t) to V? defines a continuous linear form on VI, i = 1,2. Thus we get a functional that we still label ft satisfying feeL2(0,T;Vi'). ) W > 0 (2-8) where c and a are positive constants. ') 0 , V t = l,2. 9) Moreover, as already explain, we allow also a dependence of the initial condition u0t with respect to £ and X\ but we must add the assumption K k n ( < c ( £ Q + l) W>0. ,n.

190, CH-8057 Zurich, Switzerland Abstract We study the asymptotic behaviour of the solution to lineax and nonlinear parabolic problems in cylindrical domains becoming unbounded in one or several directions. In particular we show the local stability of the solutions under changes of boundary conditions at a far away location. This generalizes a previous work in which the data depended only of the cross section of the domain. 1 Introduction In many physical situations the mathematical analysis starts after noticing that the cylindrical domain where the phenomenon is taking place being large in one direction, one can consider only what happens in a cross section reducing the difficulty by one dimension.

2 below). In fact in [2, 8, 10, 11, 13, 14, 15], the nonnegativity of the classical solutions of linear elliptic systems with nonnegative boundary data has been proved, for the class of the so-called cooperative systems. The same result was proved in [9] for parabolic systems such that the operators associated to all these equations have the same linear principal part. The nonlinear lower order terms satisfy the quasimonotonicity condition, which is the nonlinear analog of the cooperativeness condition in the linear case.

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