In this paper, we investigate the existence and uniqueness of a solution for differential equations of the carrier type on lateral boundary † of the cylinder Q, cf. (1). The main point is to transform this initial value problem into a differential operator equation of the type u 00 C M ÂZ cf. (6). The operator A is defined in Section 2, and it acts in Sobolev spaces on , boundary of . The initial value problem (6) is investigated in Section 3 by the method of Faedo-Galerkin. Thus, we obtain the existence of a weak solution for (6), and in Section 4, we prove its uniqueness.Ã Au Cˇu 0ˇ u 0 D f , and we can apply the usual methodology for the initial value problem (6).

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ResumoNeste artigo apresenta-se uma proposta de atividade tendo como objetivo utilizar a Geometria Fractal no Ensino Médio regular, onde se pode trabalhar alguns conceitos matemáticos em suas diversas áreas, tais como álgebra, cálculo, geometria plana e espacial, progressões e logaritmo de maneira diferente da convencional. Dessa forma, pretende-se reforçar a ideia atual da necessidade de experimentar a Matemática por caminhos diferentes para além da resolução de exercícios repetitivos e sem nenhum sentido lógico para o aluno. Neste sentido, a Geometria Fractal permite explorar diversos conceitos matemáticos de uma maneira mais dinâmica e criativa, por meio da construção de modelos e tabelas com os resultados das iterações, chegando a uma dedução geral do que está ocorrendo. Essa proposta é derivada de um Trabalho de Conclusão de Curso do Programa de Pós-graduação em Matemática PROFMAT, polo UNIRIO. Palavras-chave: Geometria Fractal; fractal; inovação. AbstractIn this article we present a proposal for an activity with the objective of using Fractal Geometry in regular secondary education, where we can work some mathematical concepts in its various areas, such as algebra, calculus, flat and spatial geometry, progressions and logarithm in a different way of the conventional. In this way, we intend to reinforce the current idea of the need to experiment with mathematics in different ways beyond the resolution of repetitive exercises and without any logical sense for the student. In this sense, Fractal Geometry allows us to explore several mathematical concepts in a more dynamic and creative way, through the construction of models and tables with the results of the iterations, arriving at a general deduction of what is taking place. This proposal is derived from a dissertation of the PROFMAT Mathematics Graduate Program, UNIRIO.

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Abstract. In this work we study the simultaneous controllability for a system of equations that constitutes a model of dynamical elasticity for incompressible materials.