Existence Theorem

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

In this subsection we prove an existence theorem of the equilibrium problem for the plate. The problem is formulated as a variational inequality which together with (3.2), (3.5) contains full information about other boundary conditions holding on x (0, T). An exact form of these conditions is found in the next subsection. 

The goal of this subsection is to prove an existence theorem for the optimal control problem. 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

We omit the proof of the theorem since it is analogous to that of Section 3.3 and restrict ourselves to some remarks. When proving the existence theorem the following estimates are obtained ... 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

To simplify the notations we do not indicate the dependence of the solutions on the parameters s, 5. Our aim is first to prove the existence of solutions to (5.185)-(5.188) and next to justify the passage to limits as c, 5 —> 0. A priori estimates uniform with respect to s, 5 are needed to study the passage to the limits, and we shall derive all the necessary inequalities while the existence theorem is proved. 

The contact problem for a rod under creep conditions is considered in this section. Our goal is to prove an existence theorem. We use the notations of the preceding sections. For convenience, introduce the notations... 

Johnson C. (1976) Existence theorems for plasticity problems. J. Math. Pures Appl. 55, 431-444. 

The main problem relating to practical applications of the Hohenberg and Kohn theorems is obvious the theorems are existence theorems and do not give us any clues as to the calculation of the quantities involved. 

One of the popular branches of modern mathematics is the theory of difference schemes for the numerical solution of the differential equations of mathematical physics. Difference schemes are also widely used in the general theory of differential equations as an apparatus available for proving existence theorems and investigating the differential properties of solutions. 

We really need more than a mere existence theorem that a class is translatable into so that all features of C- can be modeled in C2 without loss of... 

Theorem 1. The external potential v(r) is determined, within a trivial additive constant, by the electron density p(r). (The implication of this existence theorem is that p(r) determines the wave function and therefore all electronic properties in the ground state see also Equation 4.6.)... 

Like the first Hohenberg-Kohn theorem, the preceding theorems are existence theorems they say that the shape function is enough but they do not provide any guidance for evaluating properties based on the shape function alone. Once one knows that shape functionals exist, however, there are systematic ways to construct them using, for example, the moment expansion technique [48-51]. For atomic... 

As will be developed in more detail below, the paper by Hohenberg and Kohn (1964) [7], which proved the existence theorem that the ground state energy is a functional of n(r), but now without the approximations (valid for large N) in the explicit energy functional (1), formally completed the TFD theory. The work of Kohn and Sham (1965) [8] similarly gave the formal completion of Slater s 1951 proposal. 

Within the Hohenberg-Kohn approach [17, 18], the possibility of transforming density functional theory into a theory fully equivalent to the Schrodinger equation hinges on whether the elusive universal energy functional can ever be found. Unfortunately, the Hohenberg-Kohn theorem, being just an existence theorem, does not provide any indication of how one should proceed in order to find this functional. Moreover, the contention that such a functional should exist - and that it should be the same for systems that have neither the same number of particles nor the same symmetries (for an atom, for example, those symmetries are defined by U, L, S, and the parity operator ft) -certainly opens the door to dubious speculation. 

The first theorem of Hohenberg and Kohn is an existence theorem. As such, it is provocative with potential, but altogether unhelpful in providing any indication of how to predict the density of a system. Just as with MO theory, we need a means to optimize our fundamental quantity. Hohenberg and Kohn showed in a second theorem that, also just as with MO theory, the density obeys a variational principle. 

This theorem has immediate application to Example B above since it shows that perfect prediction is possible only when the covariance function p( ) = cn is such that In c(0) is not integrable (i.e., when the integral diverges to — oo). When k0(c) exists Theorem 1 implies immediately that for large N one has DN (ek, )N or In DN x Nk0. Our primary concern is with the higher order corrections to this asymptotic formula. 

We consider, first, whether it is in principle possible to control the quantum dynamics of a many-body system. The goal of such a study is the establishment of an existence theorem, for which purpose it is necessary to distinguish between complete controllability and optimal control of a system. A system is completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at some time T. A system is strongly completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at a specified time T. Optimal control theory designs a field, subject to specified constraints, that guides the evolution of an initial state of the system to be as close as possible to the desired final state at time T. 

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