2013/03/07 by Nicholas Katzourakis, Katzourakis, Nicholas
Mathematics · #35J62 #53C24 #Analysis of PDEs (math.AP) #FOS: Mathematics #Secondary 49J99 #math.AP #msc:35J47 #msc:35J62 #msc:49J99 #msc:53C24 #{Primary 35J47
paper · pdf · doi:10.48550/arxiv.1303.1720
5 pages, 4 figures
arxiv created 2013/03/07 · arxiv updated 2013/03/08
Given a map u : \Om \sub \Rn \larrow \RN, the ∞-Laplacian is the system \De_∞ u := (Du \ot Du + |Du|2 [Du]^\bot \ot I ) : D2 u = 0 and arises as the "Euler-Lagrange PDE" of the supremal functional E_∞(u,\Om)= ‖Du‖L^∞(\Om). \eqref1 is the model PDE of vector-valued Calculus of Variations in L^∞ and first appeared in the author's recent work \citeK1,K2,K3. Solutions to \eqref1 present a natural phase separation with qualitatively different behaviour on each phase. Moreover, on the interfaces the coefficients of \eqref1 are discontinuous. Herein we constuct new explicit smooth solutions for n=N=2 for which the interfaces have triple junctions and nonsmooth corners. The high complexity of these solutions provides further understanding of the PDE \eqref1 and shows there can be no regularity theory of interfaces.