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U δn D (F, F) As a result, we have Á µ δ Ä σ u δn C 2δkF k u δn . On the other hand, Á µ Ä σ u δn . 59) follows from the two last inequalities. 59) tends to zero as δ ! 0, n ! 1. 1 from [43] which claims that any minimizing sequence is bounded if the functional is realvalued (σ δ (u) 2 R, where R is the real number set), lower bounded (σ δ (u) > 0) and growing (σ δ (u) ! 1 as kuk ! 1). The first two properties are obvious, the third one follows from the inequality (jA δ uj, F ) Ä kA δ kkukkF k which is a consequence of the operator norm definition and the Cauchy–Bunyakovski inequality for the inner product.

9). 20) both for analytical investigation and numerical solution. Remember, that the Lagrange–Euler equation is obtained from the necessary condition of a minimum of the functional σ(u), so that its solutions are not only the minima of the functional, but also all its stationary points. 6) is investigated. In general, the minimum of σ(u) may be unattainable, that is, there may be no function u on which the minimum value of σ is reached. As will be shown later, this fact is important for theoretical investigations.

In the first step the following auxiliary problem should be solved to determine the optimal field distribution on S. Problem Va. Given desired amplitude radiation pattern F, and the shape of the exterior boundary S, find the real field distribution u 1 on S, which provides a minimum for σ t (u 1 ) D kF j f 1 jk21 C t ku 1 k2S . 100) This problem is an amplitude optimization problem; in its statement, it is similar to those for other types of antennas. 3. For the numerical algorithms and results see [21, 22].