Guido Stampacchia's paper, published in the *Annales de l’institut Fourier* in 1965, addresses the Dirichlet problem for second-order elliptic operators with discontinuous coefficients. The operator \( L \) is defined as \( Lu = -(a_{ij} u_{x_i})_{x_j} \), where \( a_{ij} \) are measurable and bounded functions satisfying an ellipticity condition. The goal is to study the Dirichlet problem for such operators, generalizing the theory of harmonic and subharmonic functions in potential theory. The paper introduces the concept of weak solutions and sub-solutions, and establishes the existence and uniqueness of solutions to the Dirichlet problem under certain conditions. Key results include the maximum principle, which states that sub-solutions satisfy \( \max_{\Omega} u \leq \max_{\partial \Omega}(\max_{\Omega} u, 0) \), and the Harnack inequality for positive solutions. The author also discusses the existence of Green's functions and their properties, and provides estimates for solutions in \( L^p(\Omega) \). The paper concludes with a detailed analysis of the regularity of solutions, showing that solutions are Hölder continuous if the coefficients satisfy additional hypotheses. The results are extended to more general operators, and the author references earlier work by other mathematicians, including W. Littman, G. Stampacchia, and H. Weinberger, and James Serrin.Guido Stampacchia's paper, published in the *Annales de l’institut Fourier* in 1965, addresses the Dirichlet problem for second-order elliptic operators with discontinuous coefficients. The operator \( L \) is defined as \( Lu = -(a_{ij} u_{x_i})_{x_j} \), where \( a_{ij} \) are measurable and bounded functions satisfying an ellipticity condition. The goal is to study the Dirichlet problem for such operators, generalizing the theory of harmonic and subharmonic functions in potential theory. The paper introduces the concept of weak solutions and sub-solutions, and establishes the existence and uniqueness of solutions to the Dirichlet problem under certain conditions. Key results include the maximum principle, which states that sub-solutions satisfy \( \max_{\Omega} u \leq \max_{\partial \Omega}(\max_{\Omega} u, 0) \), and the Harnack inequality for positive solutions. The author also discusses the existence of Green's functions and their properties, and provides estimates for solutions in \( L^p(\Omega) \). The paper concludes with a detailed analysis of the regularity of solutions, showing that solutions are Hölder continuous if the coefficients satisfy additional hypotheses. The results are extended to more general operators, and the author references earlier work by other mathematicians, including W. Littman, G. Stampacchia, and H. Weinberger, and James Serrin.