The classical Poincare Lemma states that if a smooth vector field in \(\mathbb{R}^3\) has vanishing curl, then it is the gradient of a smooth function. We give a proof, in all space dimensions, of this result when the vector field lies in a space of distributions, viz. \(H^{-1}(\Omega)\), where \(\Omega\) is a bounded open set. We also show that the Saint Venant compatibility conditions for a symmetric matrix field to be the linear strain field of a vector field resembles the Poincare Lemma at a matricial level and prove it for \(H^{-1}\) matrix fields using a similar technique. The analysis shows the intimate connection between these results and a classical lemma of Lions and Korn's inequality.
Prof. S. Kesavan is a highly respected mathematician, retired from the Institute of Mathematical Sciences (IMSc), Chennai. He completed his M.Sc. from the Indian Institute of Technology, Madras (IITM), and earned his Ph.D. from Université Pierre et Marie Curie (Paris\r\nVI). Prof. Kesavan has made significant contributions to the fields of nonlinear functional analysis, partial differential equations, and mathematical physics.
Prof. Kesavan served as a visiting and adjunct faculty member at IITM. He has authored several books, and has received recognitions for both his outstanding research and his excellence in teaching.
