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Publication# ON A GENERALIZATION OF THE POINCARE LEMMA TO EQUATIONS OF THE TYPE dw plus a Lambda w = f

Abstract

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.

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This thesis consists of two parts. The first part is about a variant of Banach's fixed point theorem and its applications to several partial differential equations (PDE's), abstractly of the form [ \mathcal Lu + \mathcal Q(u) = f.] The main result of this first part asserts that an equation having this form admits a solution if the datum $f$ satisfies a certain smallness assumption. This result (we call it \emph{the fixed point method}) is relatively simple to use and can be applied to a large variety of PDE's. The downside is that it guarantees the existence of solutions only for "small" data. The equations we deal with are Jacobian equations, non-linear elliptic PDE's, transport problems and the semi-linear wave equation. The second part of the thesis treats the $\emph{two well problem}$ in two dimensions [ \nabla u \in \mathbb{S}_A \cup \mathbb{S}_B\quad \text{almost everywhere in}\ \Omega, ] [ u = u_0\quad \text{on}\ \partial\Omega. ] For the non-degenerate case $\det(A),\det(B) \neq 0$, we show a non-existence result for piecewise regular solutions if $A$ and $B$ are non-orthogonal. For the degenerate and semi-degenerate cases, we give a characterisation for the rank-one convex hull of $\mathbb{S}_A \cup \mathbb{S}_B$ and several existence results for Lipschitz and piecewise affine solutions. Finally, for each case, we construct several explicit non-trivial solutions for well-chosen boundary conditions $u_0$.

In 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.

Assyr Abdulle, Giacomo Rosilho De Souza

Stabilized Runge???Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge???Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depends on the remaining mildly stiff components. By applying stabilized Runge???Kutta methods to this modified equation, we then devise an explicit multirate Runge???Kutta??? Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.