REDJOLA MANAJ, BESIANA ҪOBANI
Abstract
This work examines a model elliptic boundary value problem that models anisotropic diffusion inside the unit disc. Such problems arise in a wide range of physical application. The differential operator that we consider in this work includes a mixed derivative term and a coefficient which depends on the spatial variable, which makes this operator particularly interesting in the theory of elliptic partial differential equations. We begin with the weak formulation of the problem, which is studied in the Sobolev space 𝐻01(𝛺),and we analyze the associated bilinear form. By using several fundamental results from Sobolev spaces, including the Poincaré inequality, continuity of linear functionals, and the properties of uniformly elliptic matrices, we have shown that our bilinear form satisfies the condition to be both continuous and coercive. These analytical results enable the direct application of the Lax–Milgram theorem, ensuring the existence and uniqueness of a weak solution for any square-integrable source term. A complete derivation of the corresponding strong form of the partial differential equation is provided by carefully performing all integration-by-parts steps, explicitly tracking contributions from variable coefficients and mixed derivatives. The methodology presented here forms a foundation for further analytical investigations and supplies the theoretical framework necessary for numerical approximation techniques.
Key words: variational formulation, Lax-Milgram Theorem, bilinear form, well-posedness.