The boundary value problem for Yang–Mills–Higgs fields

Abstract

We show the existence of Yang–Mills–Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks–Uhlenbeck type $\alpha$-YMH fields as $\alpha \rightarrow 1$. For $\alpha >1$, some regularity results for $\alpha$-YMH field are shown. This is achieved by showing a regularity theorem for more general coupled systems, which extends the classical results of Ladyzhenskaya–Ural’ceva and Morrey.

Appendix A: Some regularity results and estimates

It is well-known that for a weakly harmonic map u, the equation of u has anti-symmetric structure $\Omega$ with $||\Omega ||_{L^2}\le C||\nabla u ||_{L^2}$ and the following regularity and estimate hold.

Lemma A.1

(see [30, Thm. 1.2]) Suppose $u\in L_1^2(D_1,{\mathbb {R}}^n)$ is a weak solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\Omega \cdot \nabla u=f\in L^p(D_1,{\mathbb {R}}^n),&{}x\in D_1\\ \frac{\partial u^a}{\partial \nu }=g^a\in L_{1,\partial }^p(\partial ^0D_1,{\mathbb {R}}^n),&{}x\in \partial ^0D_1,\quad 1\le a\le k\\ u^a=h^a\in L_{2,\partial }^p(\partial ^0D_1,{\mathbb {R}}^n),&{}x\in \partial ^0D_1,\quad k+1\le a\le n, \end{array}\right. } \end{aligned}$$

where $\Omega \in L^2(D_1,\mathfrak {so}(n)\times \wedge ^1{\mathbb {R}}^2)$, $1<p<2$ and boundary Sobolev space is defined as

$$\begin{aligned} L_{k,\partial }^p(\partial ^0D_1):=\left\{ f\in L^1(\partial ^0D_1):f={\tilde{f}}|_{\partial ^0D_1},\,{\tilde{f}}\in L_k^p(D_1) \right\} \end{aligned}$$

with norm

$$\begin{aligned} ||f ||_{L_{k,\partial }^p(\partial ^0D_1)}:=\inf _{{\tilde{f}}\in L_k^p(D_1),{\tilde{f}}|_{\partial ^0D_1}=f}||{\tilde{f}} ||_{L_k^p(D_1)}. \end{aligned}$$

Then, $u\in L_2^p(\overline{D_{1/2}},{\mathbb {R}}^n)$ and

$$\begin{aligned} ||u ||_{L_2^p(D_{1/2},{\mathbb {R}}^n)}\le C\left( ||f ||_{L^p(D_1,{\mathbb {R}}^n)}+||g ||_{L_{1,\partial }^p(\partial ^0D_1,{\mathbb {R}}^n)}+ ||h ||_{L_{2,\partial }^p(\partial ^0D_1,{\mathbb {R}}^n)}+||u ||_{L^1(D_1,{\mathbb {R}}^n)}\right) , \end{aligned}$$

provided that $||\Omega ||_{L^2(D_1)}\le \eta _0=\eta _0(p,n)$.