Summary: We consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries \(\Gamma\), with the integral taken over the domain D. We prove that the solution is a biharmonic function in \(D\) except on the interior boundaries \(\Gamma\), and satisfies some matching conditions on \(\Gamma\). There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of \((\Delta^{q}f)^{2}\), \(q\) a positive integer, then the solution is a polyharmonic function of order \(2q, \Delta^{2q}f(x) = 0,\) for \(x \in D\setminus \Gamma\), satisfying matching conditions on \(\Gamma\), and is called a polyspline of order \(2q\). Uniqueness and existence for polysplines of order \(2q\), provided that the interior boundaries \(\Gamma\) are sufficiently smooth surfaces and \(\partial D \subseteq \Gamma\), is proved. Three examples of data sets \(\Gamma\) possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional \(L\)-splines.