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Topics in partial differential equations and the calculus of variations.

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In this thesis, we study separately the topics in two aspects of the modern analysis: Partial Differential Equations and the Calculus of Variations.; In Part I, we study the differential equation {dollar}P(D)u=f{dollar} on the whole space and the exterior of an infinite cylinder in the case when f has a support inside an infinite (larger) circular cylinder. We prove that under certain conditions the solution u must have its support in the same (larger) cylinder. The main methods of our study are to use the continuation theory for the holomorphic Fourier-Laplace transform in the certain domains of several complex variables and the general idea of limiting absorption principles, along with some spectral analysis for the differential operators depending analytically on the parameter defined on the certain function spaces.; In Part II, we study the Young measure theory from the functional analysis view point and discuss some of the results related to this theory in the calculus of variations. We study the generalized Young measures determined by a sequence of gradients. We establish the lower semi-continuity theorem and the relaxation principle for the integral functional defined on the Sobolev space with a more general assumption on the integrands. Also, we prove the existence of generalized Young measure solutions. As a consequence we also derive the Euler-Lagrange equation for a generalized Young measure solution. Some of the results has played an important role in the theory of nonlinear elasticity.
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