Teoremi di confronto e di separazione per equazioni differenziali alle derivate parziali in uno spazio di Hilbert

Ravaglia, Carlo

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 58 (1975) no. 5, pp. 675-679

Résumé

The classical Sturm comparison and separation theorems have been generalized by M. Picone and by G. Cimmino to partial differential equations and to systems of ordinary and partial differential equations [1]. In [2] G. Cimmino has proved the Sturm theorems for mappings valued in $l^{2}$. Recently several authors [3, 4, 5, 6, 7] have studied the Sturm theorems for ordinary and partial equations with solutions taking their values in $\mathbf{R}$ or $\mathbf{R}^{n}$. C. Pontini [8] has proved a Sturm theorem for ordinary equations whose solutions take their values in a Banach algebra. In my work I prove Sturm comparison and separation theorems for linear homogenous selfadjoint partial differential equations of the second order whose solutions take their values in a Hilbert space. I suppose that a non-trivial solution of a differential equation taking its values in a Hilbert space vanishes on the boundary of a smooth domain $D \subset \mathbf{R}^{n}$ and I prove that a solution of another related differential equation with solutions taking their values in $\mathcal{L}(H)$ has not an inverse in at least a point of $D$. Fundamental theorems are (2) and (3) from which we obtain the identity (4) which is the basis for Sturm theorems.