THE EXISTENCE OF WEAK SOLUTIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS VIA THE LAX–MILGRAM THEOREM
Abstract
This paper addresses the theory of weak solutions for linear elliptic partial differential equations (PDEs), applying the Lax–Milgram Theorem as a central tool of Functional Analysis. In view of the restrictions imposed by classical formulations, it becomes natural to resort to Lebesgue and Sobolev spaces, in which solutions are understood in a weak sense, enabling the study of problems whose derivatives do not exist in the classical sense. We present the basic concepts of the involved function spaces, with emphasis on Hilbert and Sobolev spaces and on the Poincaré inequality, which is essential to obtain coercivity of the bilinear forms associated with elliptic operators. Based on the Riesz Representation Theorem, we prove the Lax–Milgram Theorem and analyze the continuity and coercivity conditions required for the existence and uniqueness of weak solutions in variational problems. The methodology used consisted of a bibliographic review with a deductive approach, structured in three stages: study of function spaces; Poincaré inequality and variational formulation; and the use of the theorem to characterize well-posed problems in the sense of Hadamard. A priori estimates, which are fundamental for the control and stability of solutions, are also obtained. As a result, it was verified that the Lax–Milgram Theorem constitutes a rigorous method to guarantee the existence and uniqueness of weak solutions for linear elliptic PDEs. Its application to the Poisson problem confirmed the effectiveness of the variational formulation and functional methods in the analysis of PDEs.
Author Biography
Ph.D. in Pedagogical Sciences from the Universidade de Ciências Pedagógicas Enrique José Varona (UCPEJV), Cuba. Area of specialization: Mathematical Analysis with Integration of Information and Communication Technologies. Master’s degree in Pure Mathematics from the Universidade de Cabo Verde, Faculty of Science and Technology (FCT), in partnership with the Universidade de Coimbra, Portugal. Area of specialization: Partial Differential Equations. University professor and Head of the Department of Natural Sciences and Exact Sciences at the Instituto Superior de Ciências de Educação de Cabinda (ISCED) of Cabinda, Angola.
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