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Person# David Valentin Strütt

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MATH-101(f): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

We study the system of linear partial differential equations given by dw + a Lambda w = f, on open subsets of R-n, together with the algebraic equation da Lambda u = beta, where a is a given 1-form, f is a given (k + 1)-form, beta is a given k + 2-form, w and u are unknown k-forms. We show that if rank[da] >= 2(k+1) those equations have at most one solution, if rank[da] equivalent to 2m >= 2(k + 2) they are equivalent with beta = df + a Lambda f and if rank[da] equivalent to 2m >= 2(n - k) the first equation always admits a solution. Moreover, the differential equation is closely linked to the Poincare lemma. Nevertheless, as soon as a is nonexact, the addition of the term a Lambda w drastically changes the problem.

2018In this thesis, we study two distinct problems.
The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system.
The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

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