Elastic Herglotz functions in the plane
Juan Antonio Barceló, Magali Folch-Gabayet, Salvador Pérez‐Esteva, Alberto Ruiz, M. C. Vilela · Communications on Pure & Applied Analysis · 2010
We study spaces of solutions of the spectral Navier equation in the plane.We characterize the elastic Herglotz wave functions, namely the entiresolutions $\mathbf{u}$ of the Navier equation with $L^2$far-field-patterns. The characterization is in terms of a weighted $L^2$norm involving $\mathbf{u}$ and its angular derivative $\partial_\theta\mathbf{u.}$ With respect to this norm, the space of elastic Herglotz wavefunctions is decomposed into the topological product of the compressionaland shear elastic Herglotz wave functions. We also study the solutions ofthe Navier equation whose Lamé potentials are the Fourier transform ofdistributions in the circle. We prove that these are the entire solutions ofthe Navier equation with polynomial growth. This extends a result by Agmonfor the Helmholtz equation.