Condensers with Touching Plates and Constrained Minimum Riesz and Green Energy Problems

Abstract

We study minimum energy problems relative to the \(\alpha \)-Riesz kernel \(|x-y|^{\alpha -n}\), \(\alpha \in (0,2]\), over signed Radon measures \(\mu \) on \({\mathbb {R}}^n\), \(n\geqslant 3\), associated with a generalized condenser \((A_1,A_2)\), where \(A_1\) is a relatively closed subset of a domain D and \(A_2={\mathbb {R}}^n\setminus D\). We show that although \(A_2\cap {\mathrm {Cl}}_{{\mathbb {R}}^n}A_1\) may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to \(\mu \) with \(\mu ^+\leqslant \xi \), where a constraint \(\xi \) is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted \(\alpha \)-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum \(\alpha \)-Riesz energy problem over signed measures associated with \((A_1,A_2)\) and the constrained minimum \(\alpha \)-Green energy problem over positive measures carried by \(A_1\). The results are illustrated by examples.

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Notes

  1. 1.

    See Section 3 for the notion of \(\alpha \)-thinness at infinity. The uniqueness of a solution \(\lambda _{{\mathbf {A}}}^\xi \) can be established by standard methods based on the convexity of the class of admissible measures and the pre-Hilbert structure on the linear space of all (signed) Radon measures on \({\mathbb {R}}^n\) with \(E_{\kappa _\alpha }(\mu )<\infty \), see Lemma 4.6.

  2. 2.

    When speaking of a continuous numerical function, we understand that the values are finite real numbers.

  3. 3.

    When introducing notation about numerical quantities, we always assume the corresponding object on the right to be well defined (as a finite real number or \(\pm \infty \)).

  4. 4.

    It follows from Theorem 2.4 that for a perfect kernel, such a vague cluster point exists and is unique.

  5. 5.

    In the literature, the integral representation (3.3) seems to have been more or less taken for granted, though it has been pointed out in [5, Chapter V, Section 3, \(\hbox {n}^\circ \) 1] that it requires that the family \((\varepsilon _y')_{y\in D}\) be \(\mu \)-adequate in the sense of [5, Chapter V, Section 3, Definition 1]; see also counterexamples (without \(\mu \)-adequacy) in Exercises 1 and 2 at the end of that section. A proof of this adequacy has therefore been given in [16, Lemma 3.16].

  6. 6.

    In general, \(\nu ^{D^c}({\mathbb {R}}^n)\leqslant \nu ({\mathbb {R}}^n)\) for every \(\nu \in {\mathfrak {M}}^+({\mathbb {R}}^n)\) [16, Theorem 3.11].

  7. 7.

    If Q is a given subset of D, then any assertion involving a variable point holds n.e. on Q if and only if it holds \(c_g\)-n.e. on Q, see [10, Lemma 2.6].

  8. 8.

    The strict inequality in (3.9) is caused by our convention that \(c_\alpha (D^c)>0\).

  9. 9.

    If the measure in question is positive, then Lemma 3.6 can be generalized to any bounded \(\mu \in {\mathfrak {M}}^+(D)\) such that the Euclidean distance between \(S^\mu _D\) and \(\partial D\) is \({}>0\), see [17, Lemma 3.4].

  10. 10.

    The notion of generalized condenser thus defined differs from that introduced in our recent work [11]; cf. Remark 6.7 below.

  11. 11.

    In the case of the \(\alpha \)-Riesz kernels of order \(1<\alpha \leqslant 2\) on \({\mathbb {R}}^3\), some of the (theoretical) results on the solvability or unsolvability of the condenser problem, mentioned in [24], have been illustrated in [18, 20] by means of numerical experiments.

  12. 12.

    If (4.11) is fulfilled, then \(G_{\alpha ,f}^{\xi }({\mathbf {A}},{\mathbf {1}};{\mathbb {R}}^n)\) is actually finite, see (4.9).

  13. 13.

    In Case I, the assumption of the lower boundedness of f on \(A_1\) is automatically fulfilled. Furthermore, in Case I, relation (6.3) is equivalent to the following apparently stronger assertion: \(W^\lambda _{\alpha ,f}\leqslant c\) on \(S_D^{\lambda ^+}\).

  14. 14.

    We have brought here this proof, since we did not find a reference for this possibly known assertion.

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Acknowledgements

The authors express their sincere gratitude to the anonymous referees for valuable suggestions, helping us improve the exposition of the paper.

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Correspondence to N. Zorii.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of the first author was supported, in part, by a Simons Foundation Grant No.  282207. The research of the third and the fourth authors was supported, in part, by the U. S. National Science Foundation under Grants DMS-1516400. The research of the fifth author was supported, in part, by the Scholar-in-Residence program at Purdue University Fort Wayne and by the Department of Mathematical Sciences of the University of Copenhagen.

Communicated by Doron S. Lubinsky.

10. Appendix

10. Appendix

The following example shows that even for positive bounded (hence extendible) measures on an open ball in \({\mathbb {R}}^3\), the finiteness of the \(\alpha \)-Green energy does not necessarily imply the finiteness of the \(\alpha \)-Riesz energy, contrary to what was stated in [10, Lemma 2.4].

Example 10.1

Let \(\alpha =2\). For technical simplicity, we first construct the analogous example with the ball replaced by the half-space \(D=\{(x_1,x_2,x_3)\in {\mathbb {R}}^3:\ x_1>0\}\) (next we apply a Kelvin transformation). The boundary \(\partial D\) (replacing the sphere) is then the plane \(\{x_1=0\}\). For \(r>0\), let \(\mu _r\) denote the \(\kappa _2\)-capacitary measure on the closed 2-dimensional disc \(K_r\subset \partial D\) of radius r centered at (0, 0, 0), see Remark 2.6. Such \(\mu _r\) exists since \(0<c_2(K_r)<\infty \) (in fact \(c_2(K_r)=2r/\pi ^2\), see [19, Chapter II, Section 3, \(\hbox {n}^\circ \) 14]). The Newtonian energy \(E_2(\mu _r)\) equals \(E_2(\mu _1)/r\), where \(0<E_2(\mu _1)=1/c_2(K_1)<\infty \). For real numbers \(z_1\) and \(z_2\) and a measure \(\nu \in {\mathfrak {M}}^+(\partial D;{\mathbb {R}}^3)\), denote by \(\nu ^{z_1,z_2}\) the translation of \(\nu \) in \({\mathbb {R}}^3\) by the vector \((0,z_1,z_2)\). Then \(\mu _r^{z_1,z_2}\) is the \(\kappa _2\)-capacitary measure on the translation of the disk \(K_r\) by the vector \((0,z_1,z_2)\), denoted by \(K_r^{z_1,z_2}\).

For fixed \(r>0\), the potential \(U_2^{\mu _r}\) on \({\mathbb {R}}^3\) equals 1 on the disc \(K_r\) by the Wiener criterion. By the continuity principle [19, Theorem 1.7], \(U_2^{\mu _r}\) is (finitely) continuous on \({\mathbb {R}}^3\), and even uniformly since \(U_2^{\mu _r}(x)\rightarrow 0\) uniformly as \(|x|\rightarrow \infty \), \(S_{{\mathbb {R}}^3}^{\mu _r}\) being compact (actually, \(S_{{\mathbb {R}}^3}^{\mu _r}=K_r\)).

For any positive measure \(\nu \) on \({\mathbb {R}}^3\), we denote by \((\nu )\,\check{}\) the image of \(\nu \) under the reflection \((x_1,x_2,x_3)\mapsto (-x_1,x_2,x_3)\) with respect to \(\partial D\). The 2-Green kernel \(g=g^2_D\) on the half-space D is given by

$$\begin{aligned} g(x,y)=U_2^{\varepsilon _y}(x)-U_2^{(\varepsilon _y)\,\breve{}}(x)\text { for all }x,y\in D, \end{aligned}$$

see, e.g., [1, Theorem 4.1.6], and we therefore obtain

$$\begin{aligned} E_g(\mu _r^{\varepsilon ,0})&=\int U_g^{\mu _r^{\varepsilon ,0}}\,d\mu _r^{\varepsilon ,0}=\int \Bigl (\,U_2^{\mu _r^{\varepsilon ,0}}- U_2^{(\mu _r^{\varepsilon ,0})\,\breve{}}\,\Bigr )\,d\mu _r^{\varepsilon ,0}\nonumber \\ {}&=\int U_2^{\mu _r}\,d\mu _r-\int U_2^{\mu _r}(-2\varepsilon ,x_2,x_3)\,d\mu _r(x_1,x_2,x_3)\rightarrow 0 \end{aligned}$$
(10.1)

as \(\varepsilon \downarrow 0\), noting that \(U_2^{\mu _r}(-2\varepsilon ,x_2,x_3)\rightarrow U_2^{\mu _r}(0,x_2,x_3)\) uniformly with respect to \((0,x_2,x_3)\in K_r\) as \(\varepsilon \downarrow 0\).

Consider decreasing sequences \(\{c_k\}_{k\in {\mathbb {N}}}\) and \(\{r_k\}_{k\in {\mathbb {N}}}\) of the numbers \(c_k=2^{-k}\) and \(r_k=2^{-2k}\). Then \(c_k^2/r_k=1\); hence

$$\begin{aligned} \sum _{k\in {\mathbb {N}}}\,c_k=1\text { and } \sum _{k\in {\mathbb {N}}}\,c_k^2/r_k=\infty . \end{aligned}$$
(10.2)

For \(k\in {\mathbb {N}}\), choose \(0<\varepsilon _k<1\) small enough so that

$$\begin{aligned} \bigl \Vert \mu _{r_k}^{\varepsilon _k,k}\bigr \Vert _g= \bigl \Vert \mu _{r_k}^{\varepsilon _k,0}\bigr \Vert _g<1, \end{aligned}$$
(10.3)

which is possible in view of (10.1). Now define the functional

$$\begin{aligned} \mu (\varphi ):=\sum _{k\in {\mathbb {N}}}\,c_k\mu _{r_k}^{\varepsilon _k,k}(\varphi ) \text { for all }\varphi \in C_0(D). \end{aligned}$$

Since any compact subset of D has points in common with only finitely many (disjoint) disks \(K_{r_k}^{\varepsilon _k,k}\), \(\mu \) thus defined is a positive Radon measure on D with \(\mu (D)=1\), see the former equality in (10.2). Furthermore, the partial sums

$$\begin{aligned} \eta _\ell :=\sum _{k=1}^{\ell }\,c_k\mu _{r_k}^{\varepsilon _k,k},\text { where }\ell \in {\mathbb {N}}, \end{aligned}$$

belong to \({\mathcal {E}}^+_g(D)\) with \(\Vert \eta _\ell \Vert _g<1\), the latter being clear from (10.3) and the former equality in (10.2) in view of the triangle inequality in \({\mathcal {E}}_g(D)\). Since \(\eta _\ell \rightarrow \mu \) vaguely in \({\mathfrak {M}}^+(D)\), hence \(\eta _\ell \otimes \eta _\ell \rightarrow \mu \otimes \mu \) vaguely in \({\mathfrak {M}}^+(D\times D)\) [5, Chapter III, Section 5, Exercise 5], we obtain \(\Vert \mu \Vert _g\leqslant 1\) from Lemma 2.1 with \(X=D\times D\) and \(\psi =g\).

On the other hand, being bounded, \(\mu \) is extendible to a positive Radon measure on \({\mathbb {R}}^3\) and

$$\begin{aligned} E_2(\mu )\geqslant \sum _{k\in {\mathbb {N}}}\,E_2\bigl (c_k\mu _{r_k}^{\varepsilon _k,k}\bigr )= \sum _{k\in {\mathbb {N}}}\,c_k^2E_2(\mu _{r_k})=\sum _{k\in {\mathbb {N}}}\,c_k^2r_k^{-1}E_2(\mu _1)=\infty , \end{aligned}$$

where the last equality follows from the latter equality in (10.2). This verifies Example 10.1 for a half-space.

For treating the ball, apply the inversion relative to the sphere with center (2, 0, 0) and radius 2. It maps the above half-space D on the ball \(D^*\) centered at (1, 0, 0) and with radius 1. The above measure \(\mu \) has bounded Newtonian potential \(U_2^\mu \) at the point (2, 0, 0) because \(\mu \) is bounded and supported by the closed strip \(\{0\leqslant x_1\leqslant 1\}\) not containing (2, 0, 0). Therefore, the Kelvin transform \(\mu ^*\) of \(\mu \) is a bounded measure, see [19, Eq. (4.5.3)], and can be written in the form

$$\begin{aligned} \mu ^*=\sum _{k\in {\mathbb {N}}}\,c_k\bigl (\mu _{r_k}^{\varepsilon _k,k}\bigr )^*, \end{aligned}$$

the Kelvin transformation of positive measures being clearly countably additive. Since \(\kappa _2\)-energy is preserved by Kelvin transformation, so is \(g_D^2\)-energy of the measure \(\mu _{r_k}^{\varepsilon _k,k}\in {\mathcal {E}}_2^+(D)\), as seen by combining [19, Eqs. (4.5.2), (4.5.4)] and (3.8) above. Denoting by \(g^*\) the Green kernel for the above ball \(D^*,\) we therefore obtain by (10.3),

$$\begin{aligned} \Vert \mu ^*\Vert _{g^*}\leqslant \sum _{k\in {\mathbb {N}}}\,c_k\bigl \Vert \bigl (\mu _{r_k}^{\varepsilon _k,k}\bigr )^*\bigr \Vert _{g^*}= \sum _{k\in {\mathbb {N}}}\,c_k\bigl \Vert \mu _{r_k}^{\varepsilon _k,k}\bigr \Vert _g\leqslant 1. \end{aligned}$$

And clearly \(E_2(\mu ^*)=E_2(\mu )=\infty \). This verifies Example 10.1 also for a ball.

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Dragnev, P.D., Fuglede, B., Hardin, D.P. et al. Condensers with Touching Plates and Constrained Minimum Riesz and Green Energy Problems. Constr Approx 50, 369–401 (2019). https://doi.org/10.1007/s00365-019-09454-5

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Keywords

  • Constrained minimum energy problems
  • \(\alpha \)-Riesz kernels
  • \(\alpha \)-Green kernels
  • External fields
  • Condensers with touching plates

Mathematics Subject Classification

  • 31C15