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ETNA Electronic Transactions on Numerical Analysis. Volume 23, pp. 304-319, 2006. Copyright  2006, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu ON EXTREMAL PROBLEMS RELATED TO INVERSE BALAYAGE MARIO GÖTZ Dedicated to the memory of Professor Aurel Cornea Abstract. Suppose is a body in , is compact, and a unit measure on . Inverse balayage refers to the question of whether there exists a measure supported inside such that and produce the same electrostatic field outside . Establishing a duality principle between two extremal problems, it is shown that such an inverse balayage exists if and only if "!$#%'&)( *-,/.10325476 + denotes the electrostatic potential of . where the supremum is taken! over% all unit measures . on 8 and . admitting such an inverse balayage can be characterized by a -mean-value A consequence is that pairs 6 principle, namely, A !?>@% 9 !?>@%BAC( , :D < :/;=< < for all harmonic in and continuous up to the boundary. < In addition, two approaches for the construction of an inverse balayage related to extremal point methods are presented, and the results are applied to problems concerning the determination of restricted Chebychev constants in the theory of polynomial approximation. Key words. Logarithmic potential, Newtonian potential, balayage, inverse balayage, linear optimization, duality, Chebychev constant, extremal problem. AMS subject classifications. 31A15, 30C85, 41A17. 1. Introduction. In the classical physics interpretation, balayage refers to the process of sweeping an electrical charge or gravitational mass distribution from the inside of a body in Euclidean space to the surface or boundary of this body, such that the electrostatic or, respectively, gravitational potential outside remains unchanged. Inverse balayage relates to the inverse process posing the question which electrostatic charge or mass distribution inside the body produces a given or measured potential outside. To put this into mathematical terminology, denote by EGFIH?J@HK the logarithmic FMLONQP8K or, respectively, Newtonian kernel FLSRUT8K EFMV-JXW"KZY[N ^ \]] d c d J VfegW ]]_Q`$a8b c d d jk1l J VSeiW if L)NQPhJ if LSRmTnJ and, for each (finite Borel-) measure o , define the logarithmic or Newtonian potential by prq FMVBKsY[NutvEFMV-JXW"K1L8owFxWyK j FxV{z{| + K } j Moreover, suppose ~N open set, such that B is also | , LRP , is ajbounded the boundary of the unbounded component of | . For simplicity of presentation and formulation, assume that is regular with respect to the Dirchlet problem. For the notion of Received January 2, 2006. Accepted for publication May 2, 2006. Recommended by V. Andriyevskyy. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, Ostenstraße 26, 85071 Eichstätt, Germany (mario.goetz@ku-eichstaett.de). 304 ETNA Kent State University etna@mcs.kent.edu 305 INVERSE BALAYAGE regularity and fundamental results from classical potential theory, we refer the reader to the standard literature [5], [12], [14], [16]. j Let ;F K stand for the set of (say) unit Borel measures on | , and suppose zFMK . The balayage of to B is a unit measure )N Bal F JX1K on 1 such that pr FMVBKN pr FMVBK FxVz| j K} Such a balayage always exists and can, for instance, be characterized in the following two ways (for this and more, see [9, Sect.2], [14, IV 1], [16, II.4]): (i) Denote by gFMK the collection of all functions continuous on and harmonic in the interior . Then N Bal F JX1K is uniquely characterized by the fact that integration leaves the class iFMK invariant: t L8)N t L F zCgFMKXK} In particular, it is necessary that (1.1) ¢£ ? "¡ supp ¤ ªy« D¥ FM¦§K D¥ FM¦§KZ¨ tv L8S¨ ¢@£ ©Xsupp ¤ (ii) For (possibly signed) measures (1.2) ¬ F z5gFMKXK7} J l define the (mixed) energy p ¯ ® l J l [Y N t L (provided this quantity is well-defined). Then minimizing the energy ¬ is the unique unit measure on 1 oe JXoCe among all unit measures o on 1 . Now, suppose to the contrary, that uz°±;FM1K is given, and let ~²N´ ³µ (or ³ ). An inverse balayage to on ³ is a unit measure z¶±FM³fK such that ·N Bal F J¯1K . In other words, the potential of can as wellk be produced by the more densely concentrated charge or mass . Clearly, the collection Bal FxBJX³fK of such k inverse balayages is either empty or a convex set. However, the problem to find z Bal FM¸J¯³fK is in general ill-conditioned. Important questions are: k (i) For which pairs FM¸J¯³fK is Bal FM¸J¯³fKhN¹ ~? k (ii) Do there exist designated or canonical measures z Bal FM¸J¯³fK which are in some sense minimal (e.g., with respect to their support, i.e., mother bodies or materic bodies; see [9], [10], [11], [13])? The paper is organized as follows: In Section 2 we introduce two extremal problems related to inverse balayage and state their basic properties. Section 3 generalizes these extremal problems and proves that their discrete versions are dual to one another. The key point is an interpretation in terms of duality in classical linear optimization. Conclusions providing characterizations for the existence of an inverse balayage on a given set to a given measure are drawn in Section 4. In addition, two approaches for the construction of an inverse balayage are presented. Finally, Section 5 is devoted to relating (restricted) Chebychev constants to minimax problems from the theory of polynomial approximation. ETNA Kent State University etna@mcs.kent.edu 306 M. GÖTZ 2. Extremal problems for potentials. Let Sz{ will investigate the extremal problem º p F JX1JX³fKZY[N (2.1) FM1K be fixed. In the following, we p ? "¡ p q ©Xªy« FxWyK7eÁt L8osÂ q £» ® ¤?¼½ ¥B¾ ¿;£À and relate this to inverse balayage. R EMARK 2.1. Consider oÁN²o ¼½ , the Robin equilibrium distribution of 1 [16] [12]. By Frostman’s theorem [6], which describes the Faraday cage effect in mathematical terms, p qÃÅÄ the potential is constant on . The corresponding value Æ ¼½ is called Robin constant. Then p ? "¡ p qÃÅÄ FxWyKwNÇÆ ¼½ N t Æ ¼½ L8)N t Lo ¼½ } ¿;£À Consequently, then (2.2) In fact, if t º p F J¯BJ¯³fKZRUÈ . On the other hand, if an inverse balayage to on ³ exists, º p F JX1J¯³fKZ¨mÈn} k z Bal xF BJX³fK , then for each unit measure oiz±;FBK , p LoNut p L8o{Nut p q ? "¡ p q ? "¡ p q L R¹t ;¿ £À FMWyKÉL N ¿ £À FxWyK7} One result of this paper is that, under decent assumptions on ³ and , (2.2) actually characterizes those measures Sz{;F1K , which admit an inverse balayage on ³ (Theorem 4.1). j N OTATION 2.2. A sequence FMo-ÊyK of measures on | is said to converge to a measure o Ì o ), if in the weak-star sense (symbol: oËÊ ?Í ÊÎÏ tÐ L8oÊN tÐ Lo ` for all continuous functions Ð with compact support. k R EMARK 2.3. If ³ is compact,p then Bal FM¸J¯³fK is weak-star sequentially compact. P ROPOSITION 2.4. Suppose is continuous and let ³Ñ . There exists o z ;F1K such that º n p pn ? "¡ pnqÒ F JX1JX³fKwN ¿ £À FxWyK+eÁt Lo } Fxo+Ê¸KZmFM1K with Indeed, every weak-star point of accumulation of any sequence º p p $Í $ "¡ p ;q Ó F JX1JX³CKwN ÊÎÏ ¾ ¿£ À MF WyK+e t L8oÊ Â ` is such an extremal measure. Ê £Ì Ô o in the weak-star sense Proof. Let FMo Ê KhÇ FM1K with (2.3) and suppose o Ê e along some subsequence ÕÖØ× . Note that by Helly’s selection theorem, p such weak-star points of accumulation do always exist. Then o z FM1K and, since is continuous, (2.3) (2.4) p p ?Í L8oÊN t L8o ÔSÙ Ê ÎÏ t ` } ETNA Kent State University etna@mcs.kent.edu 307 INVERSE BALAYAGE Moreover, ?Ím$ "¡ ? "¡ p ;q Ó ? "¡ p q Ò xF WyKs¨ ¿£ À FxWyK7} ÔOÙ ÊÎÏ ¿£ À ` In fact, if (2.5) was not true, then for some ÚÛCÜÈ , some WÝzU³ and ÞÇzßÕ (2.5) large, p qÒ sufficiently p qÓ ? "¡ p q Ò ? "¡ p q Ó FxW Ý KàßÚ á P)â ¿;£À xF WyKàßÚ â ¿ £À xF WyKs¨ xF W Ý K7J contradicting the fact that by weak-star convergence p q;Ó Ê £Ô FxWyKwNÇtEFãH$JXW"K1L8o Ê e Ì tvEGFIH?JãWyK1L8o N p qÒ FMWyK pointwise º p for W outside 1 . It follows from (2.3), (2.4), and (2.5) as well as from the definiJX1JX³fK that tion of F º p p ? "¡ p q Ò F JX1JX³fKZ¨ ¿£ À xF WyK7eÁt Lo ¨ º p F J¯BJ¯³fK7} R EMARK 2.5. o is, in general, not unique. For example, consider ³ØN²ä¦ Ýå ¹ and N²æ ¢Xç , the so-called harmonic measure of ¦Ý . Note that æ ¢Xç N Bal Fè ¢Xç JX1K , where here and in what follows, è ¢ãç denotes the unit point mass in ¦Ý . Then, for all ogz{;FBK , p q p q ¢ãç p q ? "¡ p q FM¦§KwN F¦ÝKN t L8è N t L} ¢£À The approach to relate the inverse balayage problem with (2.1) is to consider a second extremal problem (2.6) è"F pn pr pn $ y¡ JX1J¯³fKZYN £» ® À ¥ éX© £ ªy« ä FxVBKe FMVBK å } ¤ ¼ ½ We will establish that the quantities (2.1) and (2.6) are dual perspectives of the same thing. p J¯1JX³fKêRUÈ . R EMARKp 2.6. By the domination principle [16] [12], it is clear that è"F Moreover, è"F JX1J¯³fKNÈ if there exists an inverse balayage to on ³ . Conversely, if j 1 is also the boundary of the unbounded component of | and if there is a zFM³fK such that é ©X£ ªy¼« ½ ä p FMVBK7e p FxV1K å NQÈnJ then by the maximium principle, is an inverse balayage to . On the other hand, there are p simple examples when è"F J¯BJ¯³fK=N¶È although an inverse balayage to on ³ does not exist. For instance, this is the case when is the balayage to B of some Dirac measure è ¢ ³ , where ¦ is a point of accumulation of ³ and ³ is not ”vast” enough to carry in ¦5z an inverse balayage to . This is the reason why, henceforth, additional assumptions need to be imposed on the set ³ . P ROPOSITION 2.7. Suppose ³ë is compact. There exists z;F³fK such that è"F p J¯1JX³fKwNìé©X£ ªy« ¼½fí p Ò FMVBKwe p FxV1KîQ} ETNA Kent State University etna@mcs.kent.edu 308 M. GÖTZ Indeed, every weak-star point of accumulation of any sequences è"F (2.7) p F Ê Ksm p X Ó p ?Í JX1JX³fKwN ÊÎÏ é ©X£ ªy« ä xF V1K7e FMVBK å ¼ ½ ` F³fK with is such a measure. Proof. Suppose F ÊyK{ï;F³fK satisfies (2.7), and let be any weak-star point of Ì along Õ·× . Then zð;F³fK , since ³ is compact. By the accumulation, say Ê classical Principle of Descent [16] [12], j prÒ ?ÍG? "¡ pãÓ FxV1KsR FxV1KÑFxVðz5| K7} ÔSÙ ÊÎÏ ` Therefore, p X Ó p p Ò p $Ím? "¡ xF VBKe FMVBK å Rñé©ã£ ª'« FxV1K7e FMVBKî·} ÔSÙ ÊÎÏ éX© £ ªy¼ « ½ ä ¼½ í ` p J¯1JX³fK it follows that Taking into account (2.7) and the definition of è"F prÒ pr pn J¯1JX³fKsRé©ã£ ªy« FxVBKwe FMVBK î RUè"F J¯BJ¯³fK+} ¼½ í R EMARK 2.8. Again, is — in general — not unique. Note that any inverse balayage to on ³ (provided existence) is such an extremal measure. è"F pn 3. Duality of the extremal problems. In order to derive a duality principle related to inverse balayage, we start by formulating the previously introduced extremal problems (2.1) and (2.6) in a slightly more j general way. òJ¯óvm| be (Borel-) sets, and let ôÉFIH?J@HKõY8ò´öó Ì |ø÷ðä;àrù å be a kernel, Let ~N· which is continuous on òöó as an extended real-valued function and, say, bounded from below. For (finite Borel-) measures o on ò and on ó , set ú q FxWyKZYNutvôÉFMV-JXW"KLowFMVBKûFMWfzó3K+J ú FMVBKsY[NQtüôÉFxVËJãWyKÉL FxWyKÑFMVz5ògK+} If Ð is a measurable bounded real-valued function on ò , define ? "¡ è"F Ð JãòJ¯óKsY[N £» ® ¥ ¾ éX© ªy£ þ « ä ú xF V1Ke Ð FxVBK å Â ¤?ý FÿK and º ? "¡ q F Ð JãòðJXóKsY[N q £» ã© ªy® « þ ¥B¾ ¿ £ ú xF WyK+e tÐ Lo Â } ý ¤ Fÿ K N OTATION 3.1. We call FMÿ K the dual problem associated with FMÿK . R EMARK 3.2. Note that in the subsequent statements one could assume Ð NÈ by considering ôÉFxVËJãWyK)YNüôÉFxVËJãWyK e Ð FMVBK in FMÿK and FMÿ K , respectively. We have chosen not to do so, because the initial formulation is closer to the questions concerning the symmetric ETNA Kent State University etna@mcs.kent.edu 309 INVERSE BALAYAGE logarithmic or Newtonian kernel, and since FÿK has an interpretation as a problem of oneÐ by potentials with respect to a general kernel. Moreover, sided approximation to a function note that for ò mò and ó mó , º º è"F Ð JXò JXóKs¨mè"F Ð JXòJXó K7J T HEOREM 3.3. Suppose ò and ó º F Ð JXò JXóKZ¨ F Ð JãòJ¯ó K7} are both finite sets. Then F Ð JãòJ¯óKN·è"F Ð JãòJ¯ó3K7} Proof. W.l.o.g. we may assume that at least one of the quantities in FMÿK or FMÿ K is finite. Write òNäV J@}}@}JãV ÊBå and óüNväW J@}@}}@JXW å . The extremal problem FMÿK can be equivalently stated as the linear program Í$ H · H R RGÈ with c e Ð FxV K È .. .. .. .. .. .. . . . . . . c c J sN J N } QN e=ôÉFMV c Ê JXW K HH@He=ôÉFMV cÊ JXW K e e Ð cFxV Ê K Èc à H @ H H È È c c c c e HH@H e È È e e l Here, the first entries J@}}@}DJ of the vector z{| are the masses that a unit measure on ó associates with the points W"J}@}}DJãW , and the difference se l of two intermediate variables is the maximum of ú e Ð on ò . In linear optimization it is common to associate with the linear program FMÿwK the dual program Fÿ K given by Í ú H Hú ¨ Fÿ K ú RUÈn} This is a reformulation of FMÿ K with ú ;J@}@}}DJ ú Ê being the masses that a measure o on ò q associates with the points V;J}@}@}@JãV¸Ê and ú Ê e ú Ê l being the minimum of ú on ó . It is well-known [4, 5.2] that admissible vectors for FMÿ K and admissible ú for Fÿ K e=ôÉFMV JXW K HH@H c FMÿ K e=ôÉFMV JãWhK e are related via H R H ú } In addition, if a solution to Fÿ K (or ú to FMÿ K ) exists, then there exists also a solution ú to FMÿ K (or to FMÿ K ) and the respective objective functions are the same, i.e., Since FMÿ@K and is proved. H N H ú } FMÿ K have admissible vectors, each problem has a solution, and Theorem 3.3 ETNA Kent State University etna@mcs.kent.edu 310 M. GÖTZ Our next goal is to extend Theorem 3.3 also to non-finite sets. L EMMA 3.4 (Principle of Descent). Suppose ò and ó are compact sets. Let FMV Ê KZmò be a sequence converging to V and F Ê'KnFMó)K converging to in the weak-star sense. Then ú FMVBKs¨ ` ?Ím$ "¡ ú XÓ FxVBÊ'K+} Ê8ÎÏ Proof. The proof is classical (see [16, p.71]). From the Montone Convergence Theorem and by weak-star convergence, ?Í Í? FuJÅôÉFxVËJãWyKãKL FMWyK Î Ï t ` ?Í ? Í Í$ ?Ím$ "¡ N Î Ï ÊÎÏ t FuJ¯ôÉFMV¸ÊÉJXWyKãKÉL ÊÉFxWyKs¨ ÊÎÏ ú XÓ FxVBÊ'K+} ` ` ` L EMMA 3.5. Suppose ò and ó are compact sets and that Ð is continuous. Then º º è"F Ð JXòJXó3KwN "!©X$ªy#« è"F Ð Jãò JXóKJ F Ð JãòJ¯óKN "!©ã$ªy#« F Ð JXò JXó3K+} (3.1) ú FxV1K N "! finite %! finite Proof. First, assume to the contrary that the left-hand side equality in (3.1) does not hold. Then there exists Ú l ÜmÈ such that for all finite sets ò ò , $ y¡ ? " ¡ ú ú ! l £» ® ¤?ý ¥ é©X£ªyþ « ä FxVBKe Ð FMVBK å Ü ! £» ® ¤?ý ¥ é ©X! ªy£þ « ! ä FMV K7e Ð FxV K å àßÚ } Now, choose a sequence Fxò Ê K of finite subsets of ò Vz5ò . There are Êfz{;Fó3K with such that, say, dist FxVËJãò Ê KZ¨ c á Þ for all ? "¡ ú ú à Ú l } £» ® ?¤ ý ¥ éã© ªy£ þ« ä FMVBK7e Ð FxV1K å Ü é X©! £ ªyþ « Ó ! ä XÓ FxV Ke Ð FMV K å ø ?Í '£ & By Helly’s Selection Theorem, there exists a weak-star limit N Ê Ê z FMó)K Ç along some subsequence (UU× . Let V zò be such that ` ú ú l Ò FMV K7e Ð FxV KZÜ ©X! ªy« ! ä XÓ FxV Ke Ð FMV K å àøÚ á P (3.2) é £þ Ó and choose V Ê ziò Ê converging to V Ð is continuous on ò , . By the Principle of Descent (Lemma 3.4) and since $Ím$ y¡ ú Ó FxV Ê K+e Ð FMV Ê K å R ú Ò FxV Ke Ð FMV K7} ÊÎÏ ä &1Ù ` This contradicts (3.2). Now, assume that the second equality in (3.1) does not hold, say º º F Ð JXòJXó)KZÜm©Xþ ªy«! F Ð Jãò JXó)K-àøÚ) with Ú')nÜUÈ . Let oizFxògK be such that $ y¡ ú q $ y¡ q ! xF WyK+e tüÐ L8oßÜG©Xþ ªy«! q ! £» ©ãªy® « þ ! ¥ ¾ ¿;£ ú xF WyK+e tÐ Lo Â àøÚ')s} ¿£ ý ý ¤ ETNA Kent State University etna@mcs.kent.edu 311 INVERSE BALAYAGE There exist finite sets ò Ê ò and measures o Ê sense. Take W Ê zó such that z Fxò Ê K converging to o in the weak-star q! ? "¡ ú q xF WyK7e tÐ L8ogÜ ú Ó MF WÊyK7e tÐ L o Ê àøÚ)ê} ¿£ ý $Í W Ê ¶ z ó exists. Then by the Since ó is compact, we may w.l.o.g. assume that WQN Principle of Descent, weak-star convergence and (3.3), ` ! ú q FxWyK+eÁt Ð Loi¨ ?ÍG? "¡ ú q Ó FxW Ê Keøt Ð L8o Ê ¨ ? "¡ ú q FxWy7 K eÁt Ð LoegÚ ) J ¿£ ý ÊÎÏ ` (3.3) giving again a contradiction. L EMMA 3.6. Suppose that ò and ó that ôÉFIH?J@HK is finite on ò±öó . Then (3.4) are compact sets and that Ð is continuous. Assume ? "¡ è"F Ð JXòJXó)KwN +* !$#* è"F Ð JXòJXó K7J * ! finite º ? "¡ º F Ð JXòJXó3KN ,* !-#* F Ð JãòJ¯ó K+} * ! finite Proof. Assume contrary to the first equality in (3.4) that for some Ú.rÜUÈ , $ "¡ è"F Ð JãòðJXóKsâ * ! #* è"F Ð JãòJ¯ó K+eðÚ . } */! finite Then there exists (3.5) ó and every z MF ó)K such that for each finite set ó m ú ú ! FMVBK7e Ð xF V1K å egÚ . } é ©ãªy£þ« ä FMVBK7e Ð FxV1K å â¶é©Xªy£þ « ä z FMó K , Now, choose a sequence of finite sets ó Ê Gó and measures Ê z;Fó Ê K such that Ê in the weak-star topology. By (3.5) there exist V Ê zò with the property that (3.6) ©Xªy« ä ú FMVBKwe Ð FMVBK å â ú Ó ! FxV Ê K7e Ð FxV Ê K+eðÚ . } Ì é£þ ?Í N V Ê zÇò exists. Taking into Since ò is compact, we may w.l.o.g. assume that V Ì ú uniformly on ò (recall that ôÉFãH$JH K is assumed account that ú Ó ! finite and continuous ` on ò±öó ) as well as the continuity of Ð , we deduce ?Í ú ! 2 N ú FMV K7e Ð FxV K7J ÊÎÏ10 Ó FxV Ê K7e Ð FxV Ê K33 ` which contradicts (3.6). Thus, the first equality in (3.4) holds. Now, assume that the second equality in (3.4) does not hold, say, there exists Ú4nÜGÈ such that ? "¡ ú q $ y¡ ú q ! t Ð Â X © y ª « F x y W 7 K e L o â X © y ª « MF W K7e tÐ L8o Â egÚ 4 ! q £» ® ¤ þ ¥ ¾ ¿£ ý q ! £» ® ¤ þ ¥ ¾ ¿ £ ý ! for all finite sets ó uó . Choose a sequence ó Ê c á Þ for all Wfzó . Then there are measures o Ê z{ (3.7) uó of finite sets such that dist FxW1JXó Ê Kh¨ FMògK with $ "¡ ú q $ "¡ ú q Ó ü t Ð Â X © y ª « F M y W + K e 8 L o â MF W K7e tÐ L8oÊ3egÚ 4 } ! ¿ ; £ ¿ £ ý Ó! q £» ® ¤ þ ¥ ¾ ý ETNA Kent State University etna@mcs.kent.edu 312 M. GÖTZ By Helly’s Theorem, there exists a weak-star limit o N q;Ó Ì ú q Ò quence (ÇQ× . Since ú along ( uniformly on ó ó , we deduce that &1Ù ` ` ?Í &1Ù some subseÊÎÏ o Ê q along Ò and because ú is continuous on ?Í $ "¡ ú ;q Ó ? "¡ q Ò MF W K+e tÐ LoÊÂÁ¨ ¿£ ú xF WyK7e tÐ L8o Ê8ÎÏ ¾ ¿ ! £ ý Ó ! ý contradicting (3.7). This completes the proof of Lemma 3.6. T HEOREM 3.7. Suppose that ò and ó are compact sets and that Assume that ôÉFãH$JH K is finite on ò´ö5ó . Then J º º Ð is continuous. F Ð JãòJ¯ó3KwNQè"F Ð JXòJXóK} Proof. By Lemma 3.5, Lemma 3.6, and Theorem 3.3, º $ "¡ º $ y¡ F Ð JãòðJXóK N F Ð JXò JXó K * ! #* F Ð JXòJXó KwN * ! #* ©ã! ªy#« */! finite */! finite "! finite $ "¡ $ "¡ N *,!5#* ©ã! ªy#« è"F Ð Jãò JXó KwN *,!$#* è"F Ð JãòJ¯ó KwN¹è"F */! finite "! */! finite finite Ð JãòJ¯óK7} p ³ m and that is continuous on B . Then º pr pn F J¯1JX³fKwNQè"F JX1J¯³fK7} C OROLLARY 3.8. Suppose ³N 6 ³üNQ~ , the kernel Proof. Since 115 p ôFIH?J@HKNÇEFIH?J@HK is finite on ò´ö5ó¶NÇ1°ö5³ so . that Theorem 3.7 can be applied to Ð N It is natural to consider also a corresponding problem where the infimum and supremum in the definition of è"F Ð JãòðJXóK are exchanged. With the same approach one can deduce T HEOREM 3.9. Suppose ò and ó are both finite sets. Then ? "¡ ? "¡ ú ú q xF WyK7eÁt Ð LosÂiN Ð © X y ª « é q £» ® ¤ þ ¥ ¾ ¿ã© ª'£ ý« £» ® ¤ ý ¥¾ £ þ ä FxVBKe FMVBK å Â²} (3.8) c Outline of a proof of Theorem 3.9. Multiply both sides of (3.8) by e . Writing ò N äV J}@}@}JãV ÊBå and óN°äW J}@}@}JãW å , the linear program associated with the negative righthand side of (3.8) is Í$ H · H R7 FMÿ K RGÈ with ôFxV JãW K ÇN .. . ôÉFxVBÊÉc JãWÛ@K c e H@HHôÉFxV JXWhK .. . H@HHôÉFxVBÊÉc JãW K H@HH c H@HH e c e e .. . È È c c c .. . È È Ð FxV K JsN .. . È J Ð xF VBc ÊyK c e N .. . È à c c e } ETNA Kent State University etna@mcs.kent.edu 313 INVERSE BALAYAGE The corresponding dual program is a reformulation of the negative left-hand side of (3.8). Everything else works as in the proof of Theorem 3.3. T HEOREM 3.10. Suppose that ò and ó are compact sets and that Ð is continuous. Assume that ôÉFãH$JH K is finite on ò´ö5ó . Then ? "¡ $ y¡ ú ú q MF W"Keøt Ð L8osÂiN ã© ªy« ä FxV1K7e Ð FxVBK å Â²} q £» ® ¤ þ ¥ ¾ ¿X© ªy£ ý« £» ® ?¤ ý ¥ ¾ é£ þ Theorem 3.10 follows from Theorem 3.9 in much the same way as Theorem 3.7 follows from Theorem 3.3. We do not dwell on the proof. Reformulating the extremal problem by taking the absolute value in the definition of è"F Ð JXòJXóK one can apply the same method to obtain T HEOREM 3.11. Suppose ò and ó are both finite sets. Then $ y¡ ú q ® q ; xF WyK7e ú xF WyK7eÁt Ð L¸FMo ero l KÂ ¾ ¿£ ý dú d ? "¡ £» ® ?¤ ý ¥ ¾ é©ãªy£þ« FMVBKwe Ð FMVBK Â²} ©Xªy« 8 :® 9 8 ;<>= @® ç ? BA 8 ®@C 8;D (3.9) N Outline of a proof of Theorem 3.11. Writing òNäVÉJ}@}@}@JãV¸Ê (3.9) follows from a consideration of the linear program å and ó¶NäWÛJ@}@}}@JXW å , Í$ H · H R RGÈ with e=ôÉFMV1JXWÛDK .. . QN e=ôÉFMV¸Ê1JXWÛDK ànôÉFMV JXW K .. . ànôÉFMV c Ê JXW K c e HH@H e=ôÉFMV1JãW K .. . HH@He=ôÉFMV¸ÊÉJXW K HH@HànôÉFMV JãW hK .. . HH@H±ànôÉFMV cÊ JXW K HH@H c HH@H e c c .. .. . . c c c e c e J sN .. .. c. . c e È È È È Fÿ K e Ð FxVÉDK e .. . e Ð à Ð à Ð FxVBÊyK FxV K .. . e cFxV Ê K c È J N .. . } Èc à c e Considering the corresponding dual program, the proof of Theorem 3.11 follows the same lines as the proof of Theorem 3.3. T HEOREM 3.12. Suppose that ò and ó are compact sets and that Ð is continuous. Assume that ôÉFãH$JH K is finite on ò´ö5ó . Then ©Xªy« 8@® 9 8 ; <>=® ç ? BA 8@® C 8 ; D d d q; ? "¡ ú q ® ? "¡ MF W"Ke ú xF WyK7e tÐ LBFxoË§ero l KÂiN £» ® ¥ ¾ éX© ªy£ þ « ú FxV1K7e Ð FxVBK Âf} ¾ ¿ £ ý ?¤ ý Since Theorem 3.12 follows from Theorem 3.11 just as Theorem 3.7 follows from Theorem 3.3, we omit the proof and leave the details to the reader. ETNA Kent State University etna@mcs.kent.edu 314 M. GÖTZ 4. Characterization for the existence of an inverse balayage. Recall that, for j simpli| . For component of fication, B is assumed to be also the boundary of the unbounded p the same reason, we will impose the natural assumption that is continuous on B , and j thus in all | . p T HEOREM 4.1. Suppose ³ N ³ ë and that is continuous on 1 . Then the following statements are equivalent: k (i) Bal FxBJX³fKhNQ ~. p (ii) è"F X J 1 ¯ J f ³ K N·È . º p (iii) F JX1JX³fKwNQÈ . Proof. It remains to prove (ii) E (i). By Proposition 2.7, there exists z;F³fK such that p é ©X£ ªy¼« ½fí p Ò p FxVBKe p Ò p Ò FxVBKîN¹Èn} p Inj particular, on 1 . Now, Y[N is harmonic in | ¨ e | with non-negative boundary values on 1 . Moreover, j , continuous on \ F9ùGKN·ÈnJ if L3NÇPhJ ?Í d d jkBl xF V1KN·ÈnJ if LORUTn} F é F ÎÏ V ` j . Consequently, is an inverse balayage to . By the Minimum Principle, NÈ in | ^ _ A reformulation of (i) G (iii) is C OROLLARY 4.2. The measure with continuous potential does admit an inverse balayage on ³vN ³ m if and only if the mean-value inequality pnq ? "¡ prq FxWyKZ¨ t L8 ¿;£À holds for every measure o on 1 . Similarly, one can deduce the following consequence of Theorem 3.10. C OROLLARY 4.3. The measure with continuous potential does admit an inverse balayage on ³vN ³ m if and only if the mean-value inequality ;¿ ©ãªy£À« p q FxWyKZR t p q L8 holds for every measure o on 1 . The following corollary shows that the existence of an inverse balayage is equivalent to a mean-value inequality in the space of harmonic functions. C OROLLARY 4.4 (Mean-Value Inequality Property). The measure with continuous potential does admit an inverse balayage on ³ N ³ if and only if the mean-value inequality (4.1) ? "¡ ¿£À FMW"KZ¨ t LS¨¿©Xªy£À « FxWyK holds for every function harmonick in and continuous on . Proof. First, suppose z Bal FxBJX³fK . Then, for every zfiFMK (see (1.1)), t L)N t L $ y¡ zIH ¿£À FxWyK/J¿©Xªy£À « FxWyKKJ} ETNA Kent State University etna@mcs.kent.edu 315 INVERSE BALAYAGE Conversely, suppose that (4.1) holds, and let semi-continuous, there exists a sequence F Ê Monotone Convergence Theorem, (4.2) and in the same way, for all p q o·zG FM1K be arbitrary. p Since is lower q K²gFMK such that ÊML on 1 . By the p q $Í L ÊÎÏ t ÊhL8)N t ¦Sz5 ` , ÊÉFM¦§KwN tü ÊnLæ ¢ Ì where æ (4.3) t p q p q Læ ¢ N FM¦§K+J ¢ denotes the harmonic measure. Since ³ is compact, the latter implies that $Ím$ y¡ ? "¡ $ y¡ p q FMWyK7} ÊÎÏ ¿£À Ê FMWyKsR ¿£À ` The assertion follows from (4.1), (4.2), (4.3) and Corollary 4.2. R EMARK 4.5. Independently, Sjödin [15] has applied operator theory to the inverse balayage problem, from which an alternative proof for Corollary 4.4 can be obtained: Suppose first that the compact set ³ is so large that every function zgFK is uniquely determined by its values on ³ , e.g., if the interior of the intersection of ³ with every component of is non-empty. Despite slight modifications, the following reasoning is essentially due to [15, p.252]. If (4.1) holds, then the linear functional N satisfies dN F KwN tü L d d d F K ² ¨ é ©Xªy£À« FMVBK F zCgFMKXK F zCgFKãK7} Considering gFMK a subspace of OFM³CK , by the Hahn-Banach Theorem one may extend a linear functional on OF³fK such that dN d d d F Ð K ¨²©Xªy« Ð FxVBK F Ð zP O F³fKãKw} N to é£À By the Riesz Theorem, there exists o on ³ N with F Ð KNÇt Ð L8o F Ð zQOFM³fKXK} In particular, o is the desired inverse balayage of to ³ . Finally, in the situation of general c compact ³ one may apply the previous reasoning to ³fÊON²äVzY dist FxVËJX³fKê¨ á Þ å , let Þ tend to ù and apply an argument involving weak-star compactness (Remark 2.3), treating separately the case when some component of does not intersect with ³ . E XAMPLE 4.6. Let ¦ßzm and take ³´Nïä¦ å . By Corollary 4.4, a measure on 1 admits an inverse balayage on ä¦ å if and only if F¦ÛKwN t L for all zgFMK . This is the well-known characterization of the harmonic measure fNÇæ ¢ by the mean-value principle. R EMARK 4.7. Corollary 4.2 states that an inverse balayage to øzU±;F1K on ³N ³Ç exists if and only if for all unit measures o on 1 there exists Wzg³ such that (see (1.2)) ¬ oJãegæ ¿ U R Èn} ETNA Kent State University etna@mcs.kent.edu 316 M. GÖTZ R EMARK 4.8. (Construction of an inverse balayage). Suppose ~N ³ . The considerations in Section 3 suggest to use the following approach in order to construct an inverse balayage to given fz5±;FM1K : Step 0. Choose discretizations ò±NÇò Ê ¹1ÇóNuóR°Q³ consisting of Þ , respectively S points, such that they are asymptotically dense in the respective sets. Step 1. Solve the corresponding problem Fÿ K +(or T FMÿ UT K or FMÿ K ) of linear optimization for N FMò Ê JXó%h K. the coefficients J@}}@}@J . Note that Step 2. Consider the measures N V XT W T è X¿ Y z ;FM³CK Ì ù . If ³ is compact and if Bal k FxBJX³CKON and let ÞJ S ~ , then each weak-star point of accumulation will be an inverse balayage to on ³ . RT è¿Y If the problem FMÿ K is chosen for the optimization process, then the discrete masses will rather tend to stay away from 1 as the sets ò Ê get more dense, while the continuous formulation of the extremal problem cannot distinguish between any two inverse balayages. One may therefore be led to conjecture that the corresponding weak-star points of accumulation of the Ê are in some sense minimal inverse balayages. This should jbe compared to results of Gustafsson [11] who has shown that for convex polyhedra 1 in | and being the surface measure on B , there does exist an inverse balayage in the interior with minimal support (a so-called mother body), and it is supported by hyperplanes consisting of those points inside having at least two closest neighbours on 1 . The numerical behavior of the proposed algorithm is not considered in this paper, it will be the topic of future research. As pointed out by one of the referees, it may also be useful to incorporate a priori information of the inverse balayage measure in order to obtain good approximations in practical applications. R EMARK 4.9. A similar approach is not to associate optimal masses with given points, but given (equal) masses with optimal points: c Step 1’. Let S R and choose W§J}@}}@JãW z³ such that (4.4) \ ^ c V T p ©é X£ ªy« _ S TXW EGFxW JXVBK7e FxV1K'Z [ \ ¼½ is minimal with respect to all (not necessarily distinct) possible points on ³ . Step 2’. Consider the measures Ê N k c VÊ è ¿ Y z5 Þ XT W FM³fK7} ~ , then each weak-star point of accumulation If ³ is compact and if Bal FxBJX³CK NQ will be an inverse balayage to on ³ . T Again, the extremal problem defining the points W is such that they tend to stay away d from d 1c . Examples of Borodin [3] give further evidence for the above conjecture: Let ³Nuä ¦ ¨ T l å ¶| and suppose is the equilibrium distribution of ³ . For 1ëNv1³ , the points W coincide with the origin, a phenomenon that, by the maximum principle, can be generalized T to arbitrary lemniscatic regions. For 1¶Nu1³²l ÷gäÈ å the points W are dilated (and possibly rotated) roots of unity. Moreover, if Bü| is a smooth Jordan curve, then the quality ETNA Kent State University etna@mcs.kent.edu 317 INVERSE BALAYAGE by Ê can be quantified in relating (4.4) to the of approximation of an inverse balayage to discrepancy d d ³M] ¸JU ^ ÊU_-Y[N·©Xªy«Éä 1Fa`-K7eb ^ ÊÉFc`K Yd` any subarc of 1 å J where ^ Ê stands for the balayage of Ê to 1 (cf. [1],[2],[8]). 5. Application to polynomial approximation. Denote by e Ê FM K the set of monic c f . For h Ê zie Ê F K , (complex) polynomials of exact degree ÞR having all zeros in g let k j Ó z F K stand for the corresponding normalized zero counting measure associating c equal mass á Þ with each zero of h Ê (taking into account multiple zeros). Then polynomials are related to potentials by the simple identity p cl Ó FM¦§KwN d `?ab h c d Ê FM¦§K : m¯Ê FM¦zQfKs} òðJXó°nf and suppose o is a function positive and (say) continuous on ò . Let ~CN· D EFINITION 5.1. The number r du d ? "¡ ?Í pcq (5.1) FxòJ¯ó3KZY[N ©Xªy« Ó s Ó £'t Ó ¥ ©Xªy« ÊÉFxV1KUo Ê FMVBK ¤?ý é£þ Ê8ÎÏ ` is called restricted (weighted) Chebychev constant for the pair FxòJ¯ó3K . r ? "¡ d S q FMòJXó)KZYN ?Í ©ãªy« Ó (5.2) h ÊÉFxV1KUo Ê FxVBK j Ó £'©Xt ªyÓ « ¥ é£þ ¸ ÊÎÏ d ¤?ý ` is referred to as restricted (weighted) Maximin constant for the pair FxòðJXó3K . R EMARK 5.2. Note that (5.2) can be interpreted as a Chebychev problem for reciprocals of monic polynomials. In fact, denoting by C v Ý3w ÊÉFMó3K the set of reciprocals of monic polynomials of degree Þ with all poles in ó , c dz k Ê d ?ÍG? "¡ r Ó ? "¡ FxV1K } S q FxòJ¯ó3K N ÊÎÏ x Ó £'y ç C Ó ¤?ý ¥ é©Xªy £þ « ÊFxV1K+o ` The restricted Chebychev and Maximin constant, respectively, can be related to corresponding dual extremal problems { du dc~ Ó $Í ? "¡ pq (5.3) o L s Ó+ Á FxòJ¯ó3KZY[N ©Xªy« s Ó £'t Ó ¤ þ ¥}| ¿©ãª' £ ý« Ê-FxWyK « | Þ t ÊÎÏ ` and (5.4) S q `?ab { FxòJ¯óKZY[N ` da~ $Í Ó ? "¡ d ©Xªy« X£'© t ªy« þ | ¿£ h Ê FMWyK « | Þt j Ó Ó ¤ ¥ ý Ê8ÎÏ by means of P ROPOSITION 5.3. Suppose (5.1)-(5.4) exist and pq FxòðJXó3KwN S q ò and FMòJXó)K+J oÁL 3j Ó `$a8b ó are compact and disjoint. Then the limits in S q FxòJ¯óKN pq FMòJXó)K+} ETNA Kent State University etna@mcs.kent.edu 318 M. GÖTZ Proof. Let Ð YN o . By simple calculation, `?ab p cl Ó ? "¡ FxVBKwe Ð FMVBK å Ne (5.5) j Ó £'t Ó ¥ X© ªy« ä ?¤ ý é£ þ and `?ab d $ "¡ d Ê j Ó £'©Xt ªyÓ « ¥ é£þ h Ê FxVBK+o FxV1K ?¤ ý Xm¯Ê ? "¡ prX Ó ©XªyÓ « þ FxWyK7e tvÐ L s Ó Â ¿ £ B ¥ ¾ ý ¤ (5.6) XmXÊ du dc~ $ "¡ s Ó t N e X © y ª « É Ê x F y W K « Þ Á o L } | s Ó '£ t Ó ¤ þ } ¥ | ¿£ ý | `$a8b `?ab Now, it is not hard to see that the left-hand side in (5.5) converges to è"F Ð JXòJXóK , and the º left-hand side in (5.6) converges to F Ð JXòJXóK . By Theorem 3.7, the limits coincide, and the s Ó £'t second assertion of Proposition 5.3 is proved. The same way, the first assertion of Proposition 5.3 follows from Theorem 3.10. C OROLLARY 5.4. Suppose ò and ó are compact and disjoint. In the unweighted case c the restricted Chebychev- and Maximin-problems are asymptotically dual in the sense o that p ;FMòJXóK N S ;FMó JãòiK+} p È provided R EMARK 5.5. Suppose ò °ó . Then S FóJXògKrNÈ , while FMòJXó3KSN ò is sufficiently large, i.e., of positive logarithmic capacity. Therefore, in Proposition 5.3 the condition that ò and ó are disjoint cannot be dropped without further assumptions. On the other hand, it can be deduced from [3] and [16, Theorem III.3.1] that if ò is the closure p of a simply connected bounded domain and óNBò , then FxòJ¯ó3KfN S FMó JãògK . The corresponding relation between inverse balayage and the asymptotics of (weighted) Maximin polynomials from polynomial approximation is described in [7]. REFERENCES [1] V. V. A NDRIEVSKII AND H.-P. B LATT , Discrepancy of Signed Measures and Polynomial Approximation, Springer, New York, 2001. [2] V. V. A NDRIEVSKII , On the inverse logarithmic potential problem, East J. Approx., 7 (2001), pp. 195–204. [3] P. A. B ORODIN , On polynomials most divergent from zero on a domain boundary, Moscow Univ. Math. Bull., 52 (1997), pp. 18–21. [4] L. C OLLATZ AND W. W ETTERLING , Optimierungsaufgaben, Second ed., Springer, Berlin, 1971. [5] J.L. D OOB , Classical Potential Theory and Its Probabilistic Counterpart, Springer, New York, 1984. [6] O. F ROSTMAN , Potentiel d’équilibre et capacité des ensembles, doctoral thesis, Lund, 1935. [7] M. G ÖTZ , Maximin polynomials and inverse balayage, Comput. Methods Funct. Theory, 5 (2005), pp. 135– 141. [8] , Potential theory and discrepancy estimates in higher dimensions, in Wavelets and Allied Topics, Proceedings of a Workshop on Wavelets and Applications, P. K. Jain et al., eds., Narosa Publ., New Delhi, 2001, pp. 329–360. [9] B. G USTAFSSON , Lectures on Balayage, Clifford Algebras and Potential Theory, Univ. Joensuu Dept. Math. Rep. Ser., 7 (2004), pp. 17–63. [10] , Direct and inverse balayage — some new developments in classical potential theory, Nonlinear Anal., 30 (1997), pp. 2557–2565. [11] , On mother bodies of convex polyhedra, SIAM J. Math. Anal., 29 (1998), pp. 1106–1117. [12] L. L. H ELMS , Introduction to Potential Theory, W. de Gruyter, Berlin, 1973. [13] O. I. K OUNCHEV , Obtaining materic bodies through concentration and optimization of a linear functional, in Geophysical Data Inversion Methods and Applications, Proceedings of the 7th International Mathematical Geophysics Seminar in Berlin, Feb. 8-11 1989, A. Vogel, R. Gorenflo, C.O. Ofoegbu, and B. Ursui, eds., Friedrichs Vieweg and Son, Braunschweig, 1990, pp. 129–148. ETNA Kent State University etna@mcs.kent.edu INVERSE BALAYAGE 319 [14] N. S. L ANDKOF , Foundations of Modern Potential Theory, Springer, Berlin, 1972. [15] T. S J ÖDIN , Quadrature identities and deformation of quadrature domains, Oper. Theory Adv. Appl., 156 (2005), pp. 239–255. [16] E. B. S AFF AND V. T OTIK , Logarithmic Potentials with External Fields, Springer, Heidelberg, 1997.

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