On partially segregated harmonic maps: optimal regularity and structure of the free boundary

Abstract

We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \[\sum_{j=1}^3 \int_Ω |\nabla u_j|^2\,dx \] over a bounded domain $Ω\subset \mathbb{R}^N$, subject to the partial segregation condition: \[ u_1\,u_2\,u_3 \equiv 0 \ \text{in $Ω$.} \] We prove optimal regularity of the minimizers in spaces of Hölder continuous functions of exponent $3/4$; furthermore we prove that the free boundary is a collection of a locally finite number of smooth codimension one manifolds up to a residual set of Hausdorff dimension at most $N-2$. Finally we prove uniform-in-$β$ a priori bounds for minimal solutions to the penalized energy: \[ J_β(\mathbf{u}, Ω) = \int_Ω \sum_{i=1}^3 |\nabla u_i|^2 \,dx+ β\int_Ω \prod_{j=1}^3 u_j^2\,dx, \] in spaces of Hölder continuous functions of exponent less than $3/4$. The proofs make use of an Almgren-type monotonicity formula, blow-up analysis together with some new Liouville-type theorems.

Figures (5)

• Figure 1: In both pictures we have two triple points connected by three regular arcs. On the left, the region delimited by the two arcs of $\Gamma_{12}$ is a loop delimiting a region where both $v_1$ and $v_2$ must vanish identically. This is a not possible. On the right, the arcs of $\Gamma_{12}$ and of $\Gamma_{23}$ delimit a region where necessarily $v_2 \equiv 0$, by harmonicity. On the other hand, the two arcs of $\Gamma_{12}$ and of $\Gamma_{13}$ delimit a region where $v_1 \equiv 0$, for the same reason. As a consequence, we find an open non-empty region where two components vanishes identically, which is not possible again.
• Figure 2: The local picture and the global picture of an admissible singular point with multiplicity $2$. The numbers $1$ and $2$ identify the regions where $v_1=0$ or $v_2=0$, respectively. The connectedness of the positivity sets $\{v_i>0\}$ forces the occurrence of loops.
• Figure 3: It is not possible that two components of $\partial \omega_i$ and $\partial \omega_j$ cross each other, otherwise we always find an open non-empty region where $v_i=0$ and $v_j=0$.
• Figure 4: The two possible situations described by Case 2 in Lemma \ref{['lem: nod mer']}. We can have a finite "tower of loops", but after finitely many loops the presence of a simple triple point is necessary.
• Figure 5: On the left, the structure of the nodal set in $B_{r_0}(x_0)$. On the right, the quadruple covering on the punctured ball $B_{r_0}(x_0) \setminus \{x_0\}$, with the correct ordering of the nodal regions in order to define the function $\hat{w}$. Notice that we did not draw all the nodal lines in the second picture (for instance, there is a copy of $\Gamma_{23}$ in the first region where $\hat{w}=v_1$, but we did not draw it since it plays no role in the definition of $\hat{w}$).

Theorems & Definitions (98)

• Theorem 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1: Local Pohozaev identity
• Proposition 2.2