Boundary continuity of nonlocal minimal surfaces in domains with singularities and a problem posed by Borthagaray, Li, and Nochetto
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On the Shape of Hypersurfaces with Boundary Which Have Zero Fractional Mean Curvature
We consider compact hypersurfaces with boundary in $${\mathbb {R}}^N$$ R N that are the critical points of the fractional area introduced by Paroni et al. (Commun Pure Appl Anal 17:709–727, 2018). In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold $$\Gamma $$ Γ of dimension $$N-2$$ N - 2 and lies in a hyperplane $$H \subset {\mathbb {R}}^N$$ H ⊂ R N . Then we show that the critical points coincide with a smooth manifold $${\mathcal {N}}\subset H$$ N ⊂ H of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}= \Gamma $$ ∂ N = Γ . Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 of dimension $$N-2$$ N - 2 and suppose that $$\Gamma _1$$ Γ 1 lies in a hyperplane H perpendicular to the $$x_N$$ x N -axis and that $$\Gamma _2 = \Gamma _1 + d \, e_N$$ Γ 2 = Γ 1 + d e N ($$d >0$$ d > 0 and $$e_N = (0,\cdots ,0,1) \in {\mathbb {R}}^N$$ e N = ( 0 , ⋯ , 0 , 1 ) ∈ R N ). Then, assuming that $$\Gamma _1$$ Γ 1 has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds $${\mathcal {N}}_1 \subset H$$ N 1 ⊂ H and $${\mathcal {N}}_2 \subset H + d \, e_N$$ N 2 ⊂ H + d e N of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}_i = \Gamma _i$$ ∂ N i = Γ i for $$i \in \{1,2\}$$ i ∈ { 1 , 2 } . Moreover, the interior of the critical points does not intersect the boundary of the convex hull in $${\mathbb {R}}^N$$ R N of $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 , while this can occur in the codimension-one situation considered by Dipierro et al. (Proc Am Math Soc 150:2223–2237, 2022). We also obtain a quantitative bound which may tell us how different the critical points are from $${\mathcal {N}}_1 \cup {\mathcal {N}}_2$$ N 1 ∪ N 2 . Finally, in the same setting as in the second case, we show that, if d is sufficiently large, then the critical points are disconnected and, if d is sufficiently small, then $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 are in the same connected component of the critical points when $$N \ge 3$$ N ≥ 3 . Moreover, by computing the fractional mean curvature of a cone whose boundary is $$\Gamma _1 \cup \Gamma _2$$ Γ 1 ∪ Γ 2 , we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.
On the Shape of Hypersurfaces with Boundary Which Have Zero Fractional Mean Curvature
We consider compact hypersurfaces with boundary in $${\mathbb {R}}^N$$ R N that are the critical points of the fractional area introduced by Paroni et al. (Commun Pure Appl Anal 17:709–727, 2018). In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold $$\Gamma $$ Γ of dimension $$N-2$$ N - 2 and lies in a hyperplane $$H \subset {\mathbb {R}}^N$$ H ⊂ R N . Then we show that the critical points coincide with a smooth manifold $${\mathcal {N}}\subset H$$ N ⊂ H of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}= \Gamma $$ ∂ N = Γ . Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 of dimension $$N-2$$ N - 2 and suppose that $$\Gamma _1$$ Γ 1 lies in a hyperplane H perpendicular to the $$x_N$$ x N -axis and that $$\Gamma _2 = \Gamma _1 + d \, e_N$$ Γ 2 = Γ 1 + d e N ($$d >0$$ d > 0 and $$e_N = (0,\cdots ,0,1) \in {\mathbb {R}}^N$$ e N = ( 0 , ⋯ , 0 , 1 ) ∈ R N ). Then, assuming that $$\Gamma _1$$ Γ 1 has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds $${\mathcal {N}}_1 \subset H$$ N 1 ⊂ H and $${\mathcal {N}}_2 \subset H + d \, e_N$$ N 2 ⊂ H + d e N of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}_i = \Gamma _i$$ ∂ N i = Γ i for $$i \in \{1,2\}$$ i ∈ { 1 , 2 } . Moreover, the interior of the critical points does not intersect the boundary of the convex hull in $${\mathbb {R}}^N$$ R N of $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 , while this can occur in the codimension-one situation considered by Dipierro et al. (Proc Am Math Soc 150:2223–2237, 2022). We also obtain a quantitative bound which may tell us how different the critical points are from $${\mathcal {N}}_1 \cup {\mathcal {N}}_2$$ N 1 ∪ N 2 . Finally, in the same setting as in the second case, we show that, if d is sufficiently large, then the critical points are disconnected and, if d is sufficiently small, then $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 are in the same connected component of the critical points when $$N \ge 3$$ N ≥ 3 . Moreover, by computing the fractional mean curvature of a cone whose boundary is $$\Gamma _1 \cup \Gamma _2$$ Γ 1 ∪ Γ 2 , we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.
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