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Boundary Unique Continuation on \(C^1\)-Dini Domains and the Size of the Singular Set

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A Correction to this article was published on 16 May 2022

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Abstract

Let u be a harmonic function in a \(C^1\)-Dini domain \(D\subset {\mathbb {R}}^d\) such that u vanishes on a boundary surface ball \(\partial D\cap B_{5R}(0)\). We consider an effective version of its singular set (up to boundary) \(\mathcal {S}(u):=\{X\in {\overline{D}}: u(X) = |\nabla u(X)| = 0\} \) and give an estimate of its \((d-2)\)-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of u centered at 0. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

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Notes

  1. see Proposition 7.3 as well as Lemma 9.50. Heuristically as \(r\rightarrow 0+\), \(T_{X,r} u\) converges to a homogeneous harmonic polynomial \(P_N\) of degree \(N\in \mathbb {N}\), and it satisfies \(\iint _{B_1(0)} |P_N|^2 \,{\mathrm{d}}Z = 1\). In particular it implies that \(\iint _{B_1(0)}|\nabla P_N|^2 \,{\mathrm{d}}Z = 2N+d\). In the special case when \(N=1\) and \(P_N\) is linear, \(|\nabla P_N|\) equals a dimensional constant, denoted by \(\alpha _d\). Here \(\alpha _0\) is chosen to be strictly smaller than \(\alpha _d\). On the other hand when the degree \(N>1\), by homogeneity \(|\nabla P_N(r\omega )| = O(r^{N-1}) \) grows polynomially in the radial direction. Since \(\iint _{B_1(0)}|\nabla P_N|^2 \,{\mathrm{d}}Z = 2N+d\) and it has a uniform upper bound when \(N\leqq C(\Lambda )\), the polynomial growth of \(|\nabla P_N|\) implies that we can choose \(\beta <1\) so that for any degree \(N\leqq C(\Lambda )\), \(\sup _{B_\beta (0)} |\nabla P_N|\) is also strictly smaller than \(\alpha _d\). (It’s certainly not the case that \( |\nabla P_N| < \alpha _d\) on all of \(B_1(0)\).)

  2. This can be made rigorous in the interior case. By [16, Theorem 3.1] the harmonic function has an expansion of the form \(u(X+Y) - u(X) = P_N(Y) + \Psi (Y)\), where \(P_N\) is a homogeneous harmonic polynomial of degree \(N \in {\mathbb {N}}\) and \(\Psi (Y) = O(|Y|^{N+\epsilon })\); moreover \(\Vert \nabla T_{X,r} u - \nabla P_N\Vert _{L^p(B_1(0))} = O(r^{\epsilon + \frac{d}{p} - 1})\rightarrow 0\) for any \(p\in (1, d]\).

  3. For any \(X\in {\mathbb {R}}^d\), we say h is N-homogeneous with respect to X if \(h-h(X)\) is N-homogeneous, namely \(h(X+\lambda Z) - h(X) = \lambda ^{N}( h(X+Z) - h(X))\) for any \(Z\in {\mathbb {R}}^d\) and \(\lambda \in {\mathbb {R}}_+\).

  4. Let \({Y}\) be an arbitrary vector, then its projection onto the radial direction can be written as \(\left\langle {Y}, \frac{ \frac{1}{2} \nabla _g |X|^2}{|X|} \right\rangle _g = \left\langle {Y}, \frac{X}{|X|} \right\rangle \).

  5. Throughout the paper we will often require the scale to be sufficiently small. Unless otherwise specified it always means that the radius is less or equal to R satisfying the assumption here.

  6. If \(u(p) \ne 0\), then the leading order of u near p is simply the non-trivial constant u(p). In this case, we need to define the frequency function centered at p using the harmonic function \(u-u(p)\), so as to capture the leading order term of \(u-u(p)\).

  7. By Lemmas 5.1 and 6.8 we know the bound \(\Lambda _* \leqq C(\Lambda )\).

  8. If \(r_*/6 \ne r_0/\rho ^{i}\) for any integer \(i\in \mathbb {N}\), we just replace it by \(r_0/\rho ^{i_*}\) where \(i_*\) is the largest integer such \(r_0/\rho ^i \leqq r_*/6\).

  9. The only exception is that it is possible in our covering argument (at the top level) that \(0\in {{\,\mathrm{spt}\,}}{{\bar{\mu }}}\) even though \(0\not \in \mathcal {S}\). But we ignore this case since this can only happen at a single point.

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  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Carlos Kenig & Zihui Zhao

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Correspondence to Zihui Zhao.

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Communicated by F. Lin.

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The original online version of this article was revised to add references [6, 7].

The first author was supported in part by NSF Grant DMS-1800082, and the second author was partially supported by NSF Grant DMS-1902756.

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Kenig, C., Zhao, Z. Boundary Unique Continuation on \(C^1\)-Dini Domains and the Size of the Singular Set. Arch Rational Mech Anal 245, 1–88 (2022). https://doi.org/10.1007/s00205-022-01771-7

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