Multiple integrals in the calculus of variations and nonlinear elliptic systems

Weakly Differentiable Functions

Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ)

We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq \Lambda$. Assume that $\Sigma_k$ converges to $\Sigma$ in moduli space, i.e. $\phi_k^\ast(\Sigma_k) \to \Sigma$ as complex structures for diffeomorphisms $\phi_k$. Then we construct a branched conformal immersion $f:\Sigma \to \R^n$ and M\"obius transformations $\sigma_k$, such that for a subsequence $\sigma_k \circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $\Lambda < 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k) \geq \min(8\pi,\omega^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2012/0020/0002/CAG-2012-0020-0002-a004.pdf

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Papers by: S.H. Abed, O. Amici, A. Asada, V.V. Astrhancev, S. Bacso, J.K. Beem, A. Bejancu, K. Belteky, I.M. Benn, L. Biesk, T.Q. Binh, N. Blazic, N. Bokan, K. Buchner, B. Casciaro, A.V. Chakmazjan, I. Comic, A. Crumeyrolle, I. Dimitric, C.T.J. Dodson, P. Dragila, S. Dragomir, J.J. Duistermaat, P.E. Ehrlich, J. Eichhorn, M. Falcitelli, M. Ferraris, S. Formella, M. Francaviglia, S.I. Goldberg, A. Gray, T. Hangan, M. Husty, G. Ivan, A. Jakubowicz, S. Janeczko, Z. Jankovic, J. Klein, F.C. Klepp, I. Kolar, Z. Kovacs, L. Kozma, B. Lukacs, V.S. Malakhovsky, K.B. Marathe, S. Markvorsen, I. Mihai, J. Mikesh, R. Miron, E. Molnar, A.M. Naveira, J. Nikic, J. Novotny, V. Oliker, T. Otsuki, A.M. Pastore, Z. Perjes, J.F. Pommaret, M. Puta, H. Reckziegel, A.H. Rocamora, O. Roschel, H. Sachs, A. Sebestyen, U. Simon, H. Singh, G.A.J. Sparling, P. Stavre, S. Steiner, R. Sulanke, J. Szenthe, J. Szilasi, L. Tamassy, E. Teufel, F. Tricerri, R.W. Tucker, C. Udriste, I. Vaisman, L. Vanhecke, L. Verstraelen, V.A. Vujicic, P.G. Walczak, H. Yasuda, A. Zajtz, V. Zoller

Topics in Differential Geometry

We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity We find conditions that are both necessary and sufficient for the compactness in W 1,2 and C 0 modulo bubbles of sequences of such maps

/pdf/harmonic-maps-from-degenerating-riemann-surfaces-u3pbycj8nx.pdf

Harmonic maps from degenerating Riemann surfaces

Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We show that a weakly Diracharmonic map is smooth in the interior of the domain. We also prove regularity results for Dirac-harmonic maps at the boundary when they solve an appropriate boundary value problem which is the mathematical interpretation of the D-branes of superstring theory.

/pdf/the-boundary-value-problem-for-dirac-harmonic-maps-15zxyvgt2i.pdf

The boundary value problem for Dirac-harmonic maps

Multi-hierarchical nanofiber membrane with typical curved-ribbon structure fabricated by green electrospinning for efficient, breathable and sustainable air filtration

We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface ${N \subset \mathbb{R}^{d+1}}$ is smooth.

/pdf/regularity-for-weakly-dirac-harmonic-maps-to-hypersurfaces-1clmpj3cfe.pdf

Regularity for weakly Dirac-harmonic maps to hypersurfaces

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $\epsilon $-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.

/pdf/regularity-at-the-free-boundary-for-dirac-harmonic-maps-from-4ffdx1pra4.pdf

Miaomiao Zhu

Papers

Harmonic maps from degenerating Riemann surfaces

The boundary value problem for Dirac-harmonic maps

Multi-hierarchical nanofiber membrane with typical curved-ribbon structure fabricated by green electrospinning for efficient, breathable and sustainable air filtration

Regularity for weakly Dirac-harmonic maps to hypersurfaces

Regularity at the free boundary for Dirac-harmonic maps from surfaces