Generalized functions on the closure of an open set. Application to uniqueness of some characteristic Cauchy problem

Victor Dévoué, Jean-André Marti, Hans Vernaeve, Jasson Vindas

To obtain existence and uniqueness when solving some nonlinear characteristic Cauchy problems, we define a special algebra $\mathcal{G}_{\mathcal{O}_{M}% }\left( \overline{\Omega}\right) $ of generalized functions on the closure $\overline{\Omega}$ of an open set $\Omega$ in $\mathbb{R}^{n}$ constructed from the topological algebra $\mathcal{O}_{M}\left( \overline{\Omega}\right) $ of slowly increasing functions in $\overline{\Omega}$. Moreover other concepts are needed as slow scale elements and point values characterization of elements in $\mathcal{G}_{\mathcal{O}_{M}}\left( \overline{\Omega}\right) $.