We present the definitions, derive the relevant Euler-Lagrange equations, and establish various properties concerning biharmonic maps. We investigate several classes of examples exhibiting singular behavior. Existence of a weak solution to the associated evolution equation is proved using a penalization argument and the Galerkin method. We prove higher integrability of order greater than two for derivatives of Laplacian energy minimizers contingent upon certain energy constraints. We initiate a numerical analysis of biharmonic maps using a discrete Laplacian energy, a finite difference scheme, and involving spherical coordinates in a variety of dimensions in order to understand isolated singularities. Since singularities of Laplacian energy minimizers first occur in dimension five, where both mathematical and numerical analysis is quite complicated, we also consider an analogous problem for a lower non-integer power, p, of the Laplacian. The Euler-Lagrange equations are now not only nonlinear and fourth-order, but also degenerate. Nevertheless, we are able to analyze a specific solution of the p-biharmonic map equation having a degree two singularity. We prove the non-minimality of the solution by a construction that splits this singularity.
In this work, we have developed relevant computer code to expedite biharmonicity testing and energy computations. We have also developed computer code applicable to the initial study of the discretized biharmonic problem. We hope this code will be useful in future research aimed at a more comprehensive numerical analysis.