Minimization problem

The minimization problem (1.57) also provides the fulfilment of the equilibrium equations... 

Let us emphasize that not model can be presented as a minimization problem like (1.55) or (1.57). Thus, elastoplastic problems considered in Chapter 5 can be formulated as variational inequalities, but we do not consider any minimization problems in plasticity. In all cases, we have to study variational problems or variational inequalities. It is a principal topic of the following two sections. As for general variational principles in mechanics and physics we refer the reader to (Washizu, 1968 Chernous ko, Banichuk, 1973 Ekeland, Temam, 1976 Telega, 1987 Panagiotopoulos, 1985 Morel, Solimini, 1995). 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

Now we can prove the statement of solution existence to minimization problems. 

Then P V K is called a projection operator. Thus finding the projection (1.93) is equivalent to solving the minimization problem... 

Theorem 3.6. There exists a solution of the minimization problem... 

In this case the solution W of the minimization problem satisfies (4.101)-(4.103) and some additional boundary conditions holding on S . These conditions are analysed below (see (4.128)). 

Air pollution from this process is a minimal problem because all waste gases are burned for fuel value or flared. The products of combustion of the... 

Meissner, R. E., Ill and D. C. Shelton. 1992. An Engineer s Guide to Plant Layout. Part 1. Minimizing Problems in Plant Eayout. Chemical Engineering, 99(4), 81-85. 

It allows more flexibility to engineers with regard to the type of flame arrester nsed and its approval, dins helping to possibly minimize problems such as maintenance, plngging, and pressure drop. 

To prevent or minimize problems associated with concentration cells, the following methods should be considered in general ... 

Several Machines on the "Non-Vectorizable" Minimization Problem, and Data from the Dongarra-LINPACK Test... 

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