Interior partial regularity for minimal \(L^p\)-vector fields with integer fluxes (Q2792156)

Interesting questions arise from the problem of defining ``weak bundles'' with connections preferably in Sobolev classes. The minimization problem discussed here is inspired from the discussion of singular \(U(1)\)-bundles in [the author and \textit{T. Rivière}, Geom. Funct. Anal. 21, No. 6, 1419--1442 (2011; Zbl 1242.58009)].NEWLINENEWLINEAn accessible formulation of the problem is as follows. A vector field \(X\in L^p (B^3,\mathbb R^3)\) is called a vector field with integer fluxes if for every ball \(B_r(a)\) strictly inside \(B\), the number \(\int_{\partial B_r(a)}X\cdot\nu\) is an integer. For \(p\geq 3/2\), this is just the class of divergence-free vector fields, but here the focus is on \(1<p<3/2\). It is proven that minimizers of \(\int_B|X|^p\) among vector fields with given boundary data for \(X\cdot\nu\) and with integer fluxes are locally Hölder continuous away from a locally finite subset of \(B\).NEWLINENEWLINESome parts of the proof resemble those from harmonic map theory, using monotonicity, stationarity, and a variant of the Luckhaus Lemma for sequential compactness of minimizers. The proof of \(\varepsilon\)-regularity, however, brings in new ideas. It uses the fact that the integer fluxes condition helps identify singular points in \(L^p\)-vector fields which come with natural integer ``charges''. Approximation of vector fields by more simple ones is then possible, and the charged points are vertices of some directed graph. This allows to use combinatoric methods as well as those from geometric measure theory.