We delve into the intricate Cauchy problem entwined with the biharmonic equation, set within a confined expanse of \({\mathbb{R}}^{2}\) . Through meticulous derivation, we unveil the requisite conditions for the existence of solutions, alongside a Carleman formula that specifically pertains to this Cauchy problem in \({\mathbb{R}}^{2}\) . Our exploration reveals that this problem possesses a density of solvable instances; nevertheless, inputs presenting compact support nestled within the interior of \(S\) remain outside the solution set. This compelling observation prompts us to conclude that the problem is ill-posed, thereby challenging the efficacy of traditional Fourier integral operator techniques. Conversely, when \(S\) is characterized as real analytic, the venerable Cauchy–Kovalevskaya theorem offers a beacon of assurance, guaranteeing the existence of a local solution.
