Operational approach for biharmonic equations in $${\varvec{L}}^{{\varvec{p}}}$$-spaces

Abstract

In this work, we study the existence, uniqueness and maximal $$L^p$$-regularity of the solution of different biharmonic problems. We rewrite these problems by a fourth-order operational equation and different boundary conditions, set in a cylindrical n-dimensional spatial region $$\Omega $$ of $${\mathbb {R}}^n$$. To this end, we give an explicit representation formula, using analytic semigroups, and invert explicitly a determinant operator in $$L^p$$-spaces thanks to $$\mathcal {E}_\infty $$ functional calculus and operator sums theory.