THE EXISTENCE OF WEAK SOLUTIONS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS VIA THE LAX–MILGRAM THEOREM
Abstract
This article analyzes the theory of weak solutions for linear elliptic partial differential equations (PDEs), grounded in the Lax-Milgram Theorem a fundamental result in Functional Analysis. It addresses the limitations of classical formulations, which necessitate the transition to Lebesgue and Sobolev function spaces. In these spaces, solutions are redefined in a generalized (weak) sense, allowing the treatment of problems where derivatives do not exist in the classical sense. The core of the analysis lies in the application of this theorem in Hilbert spaces, providing conditions for the existence and uniqueness of variational problems. By verifying the continuity and coercivity of the bilinear form associated with the elliptic operator, and with the support of the Riesz Representation Theorem, the structure of the solutions is consolidated. Furthermore, the use of a priori estimates ensures control over the behavior of these solutions.
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. Professor at the Instituto Superior de Ciências de Educação de Cabinda (ISCED-CABINDA), Angola.
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