On some transmission problems set in a biological cell, analysis and resolution

Some transmission problems are set in bodies with a crown of small thickness ε>0. For instance, those concerning the conductivity in the biological cell. By a natural change of variables, we transform them in transmission problems set in two cylindrical bodies ]−∞,0[×]−π,π[ and ]0,δ[×]−π,π[ (where δ=ln⁡(1+ε)) and then, in some general elliptic abstract differential equations (Pδ)δ>0. The goal of this work is to give a complete study of these problems (Pδ)δ>0 for every δ>0. Existence, uniqueness and maximal regularity results are obtained for the classical solutions essentially by using the semigroups theory.