1615:
346:
1456:
714:
1148:
1048:
849:
2434:
2808:. A historical account of the development of capacity theory by its founder and one of the main contributors; an English translation of the title reads: "The birth of capacity theory: reflections on a personal experience".
509:
2600:
1314:
224:
1911:
2157:
2015:
1818:
2509:
1197:
584:
146:
2241:
1408:
2303:
2332:
1610:{\displaystyle C(\Sigma ,S)=-{\frac {1}{2\pi }}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma '={\frac {1}{2\pi }}\int _{D}|\nabla u|^{2}\mathrm {d} x}
2749:, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics., vol. 19 (2nd ed.), Bombay: Tata Institute of Fundamental Research,
2264:
1940:
1706:
1652:
603:
2779:
1448:
2070:
2044:
1736:
2766:. The second edition of these lecture notes, revised and enlarged with the help of S. Ramaswamy, reâtypeset, proof read once and freely available for download.
1079:
962:
747:
2340:
435:
2461:
2690:. The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.
2520:
2924:
2834:
1243:
341:{\displaystyle C(\Sigma ,S)=-{\frac {1}{(n-2)\sigma _{n}}}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma ',}
900:, the region bounded by ÎŁ, can be found by taking the condenser capacity of ÎŁ with respect to infinity. More precisely, let
2988:
2970:
2952:
2983:
2965:
2947:
1837:
2090:
1948:
2916:
1748:
3004:
2470:
739:
2449:
720:
2978:
2050:
736:
2448:
achieving particular boundary values, given above, can be extended to other energy functionals in the
1173:
560:
122:
2172:
2960:
2859:
1359:
2714:
2273:
1739:
1679:
1230:
929:
175:
2683:
2308:
371:
1153:
The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let
709:{\displaystyle C(\Sigma ,S)={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla u|^{2}\mathrm {d} x.}
2942:
2920:
2863:
2830:
2699:
2445:
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359:
48:
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1685:
2930:
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2848:
2796:
2758:
2720:
2675:
2163:
1659:
1631:
187:
59:. The potential energy is computed with respect to an idealized ground at infinity for the
56:
40:
2887:
2844:
2792:
2754:
1421:
2934:
2895:
2883:
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2800:
2788:
2770:
2762:
2750:
2625:
2055:
1058:
855:
80:
36:
2871:
2023:
1715:
1628:
arises, as the potential function goes from being an inverse power to a logarithm in the
2743:
Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.)
2909:
2726:
1143:{\displaystyle C(K)=\int _{S}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma .}
101:
2998:
2774:
2081:
163:
2775:"La naissance de la théorie des capacités: réflexion sur une expérience personnelle"
1043:{\displaystyle C(K)=\int _{\mathbb {R} ^{n}\setminus K}|\nabla u|^{2}\mathrm {d} x.}
186:− 1)-dimensional hypersurface that encloses ÎŁ: in reference to its origins in
844:{\displaystyle I={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla v|^{2}\mathrm {d} x}
416:
112:
47:
or physical extent, capacity is a mathematical analogue of a set's ability to hold
2741:
2702: â number that denotes how big a certain bounded analytic function can become
17:
2708:
2429:{\displaystyle E(\lambda )=\int \int _{K\times K}G(x-y)d\lambda (x)d\lambda (y)}
1418:
In two dimensions, the capacity is defined as above, but dropping the factor of
1320:
1065:, then the harmonic capacity can be equivalently rewritten as the integral over
554:
413:
171:
52:
28:
504:{\displaystyle {\frac {\partial u}{\partial \nu }}(x)=\nabla u(x)\cdot \nu (x)}
2822:
2780:
Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences
167:
893:
195:
109:
79:
The notion of capacity of a set and of "capacitable" set was introduced by
2872:"Regular points for elliptic equations with discontinuous coefficients"
2821:, Grundlehren der Mathematischen Wissenschaften, vol. 262, Berlinâ
2805:
2621:
732:
2915:, London Mathematical Society Student Texts, vol. 28, Cambridge:
55:
of the set: the total charge a set can hold while maintaining a given
1163:
44:
2595:{\displaystyle I=\int _{D}(\nabla u)^{T}A(\nabla u)\,\mathrm {d} x}
2444:
The characterization of the capacity of a set as the minimum of an
1319:
The harmonic capacity is a mathematically abstract version of the
2876:
Annali della Scuola
Normale Superiore di Pisa â Classe di Scienze
1229:) will form a condenser pair. The harmonic capacity is then the
2723: â Harmonic functions as solutions to Laplace's equation
2818:
Classical potential theory and its probabilistic counterpart
1654:
limit. This is articulated below. It may also be called the
2662:
The minimum energy is achieved by a function known as the
1309:{\displaystyle C(K)=\lim _{r\to \infty }C(\Sigma ,S_{r}).}
2900:
2246:
with the infimum taken over all positive Borel measures
2729: â Area of functional analysis and convex analysis
2711: â Ability of a body to store an electrical charge
39:
is a measure of the "size" of that set. Unlike, say,
2523:
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2343:
2311:
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1951:
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and is always non-negative and finite: 0 â€
1246:
1176:
1082:
965:
750:
606:
597:) can be equivalently defined by the volume integral
563:
438:
227:
125:
2704:
Pages displaying wikidata descriptions as a fallback
2514:are minimizers of the associated energy functional
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1402:
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1142:
1042:
843:
708:
578:
503:
340:
140:
2198:
2162:The variational definition of capacity over the
1263:
83:in 1950: for a detailed account, see reference (
1906:{\displaystyle G(x-y)={\frac {1}{|x-y|^{n-2}}}}
2152:{\displaystyle C(K)=\int _{S}d\mu (y)=\mu (S)}
2010:{\displaystyle G(x-y)=\log {\frac {1}{|x-y|}}}
904:be the harmonic function in the complement of
8:
2605:subject to appropriate boundary conditions.
1813:{\displaystyle u(x)=\int _{S}G(x-y)d\mu (y)}
2682:with the obstacle function provided by the
553: â 2) is the surface area of the
2504:{\displaystyle \nabla \cdot (A\nabla u)=0}
402:is any intermediate surface between ÎŁ and
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67:, and with respect to a surface for the
998:
84:
2717: â Green's function for Laplacian
2462:elliptic partial differential equation
1658:, in reference to its relation to the
2911:Potential theory in the complex plane
2825:âNew York: Springer-Verlag, pp.
2626:continuously differentiable functions
856:continuously differentiable functions
7:
1069:of the outward normal derivative of
218:), is given by the surface integral
2655:) = 0 on the boundary of
2084:. It is related to the capacity as
1203:is bounded, for sufficiently large
2585:
2574:
2552:
2486:
2474:
2456:Divergence form elliptic operators
1600:
1581:
1536:
1525:
1517:
1466:
1284:
1273:
1130:
1119:
1111:
1030:
1011:
834:
815:
719:The condenser capacity also has a
696:
677:
613:
471:
450:
442:
323:
312:
304:
234:
25:
932:of the simple layer ÎŁ. Then the
2080:. It is generally taken to be a
1192:{\displaystyle \mathbb {R} ^{n}}
579:{\displaystyle \mathbb {R} ^{n}}
141:{\displaystyle \mathbb {R} ^{n}}
2236:{\displaystyle C(K)=\left^{-1}}
2580:
2571:
2559:
2549:
2533:
2527:
2492:
2480:
2423:
2417:
2408:
2402:
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2315:
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2210:
2185:
2179:
2146:
2140:
2131:
2125:
2103:
2097:
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1986:
1967:
1955:
1884:
1869:
1856:
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1807:
1801:
1792:
1780:
1761:
1755:
1638:
1589:
1577:
1475:
1463:
1437:
1425:
1403:{\displaystyle C(K)=e^{-W(K)}}
1395:
1389:
1372:
1366:
1300:
1281:
1270:
1256:
1250:
1092:
1086:
1019:
1007:
975:
969:
823:
811:
784:
772:
760:
754:
685:
673:
646:
634:
622:
610:
498:
492:
483:
477:
465:
459:
270:
258:
243:
231:
1:
2298:{\displaystyle \lambda (K)=1}
119:-dimensional Euclidean space
51:. More precisely, it is the
2977:Solomentsev, E. D. (2001) ,
2959:Solomentsev, E. D. (2001) ,
2941:Solomentsev, E. D. (2001) ,
721:variational characterization
2984:Encyclopedia of Mathematics
2966:Encyclopedia of Mathematics
2948:Encyclopedia of Mathematics
2327:{\displaystyle E(\lambda )}
1738:. It can be obtained via a
896:, the harmonic capacity of
3021:
2917:Cambridge University Press
2907:Ransford, Thomas (1995),
2612:with respect to a domain
2460:Solutions to a uniformly
1620:This is often called the
873:) = 1 on ÎŁ and
381:) = 1 on ÎŁ and
43:, which measures a set's
2740:Brélot, Marcel (1967) ,
2259:{\displaystyle \lambda }
2166:can be re-expressed as
1059:rectifiable hypersurface
912: = 1 on ÎŁ and
174:) set of which ÎŁ is the
2624:of the energy over all
2334:is the energy integral
1935:{\displaystyle n\geq 3}
1701:{\displaystyle n\geq 3}
2608:The capacity of a set
2596:
2505:
2450:calculus of variations
2430:
2328:
2299:
2260:
2237:
2153:
2066:
2040:
2011:
1936:
1907:
1814:
1732:
1702:
1670:The harmonic function
1648:
1647:{\displaystyle n\to 2}
1611:
1444:
1404:
1321:electrostatic capacity
1310:
1193:
1144:
1044:
956:), is then defined by
924: â â. Thus
845:
710:
580:
505:
362:defined on the region
342:
142:
108: − 1)-
2597:
2506:
2464:with divergence form
2431:
2329:
2300:
2270:, normalized so that
2261:
2238:
2154:
2067:
2041:
2012:
1937:
1908:
1815:
1733:
1710:logarithmic potential
1703:
1649:
1612:
1445:
1443:{\displaystyle (n-2)}
1405:
1335:) < +â.
1311:
1194:
1145:
1061:completely enclosing
1045:
846:
711:
581:
506:
343:
143:
2979:"Energy of measures"
2674:, and it solves the
2664:capacitary potential
2521:
2471:
2341:
2309:
2274:
2250:
2173:
2091:
2065:{\displaystyle \mu }
2056:
2024:
1949:
1920:
1838:
1827:a point exterior to
1749:
1716:
1686:
1632:
1622:logarithmic capacity
1457:
1422:
1414:Logarithmic capacity
1360:
1244:
1174:
1170:about the origin in
1080:
963:
748:
604:
561:
436:
225:
190:, the pair (ÎŁ,
123:
2715:Newtonian potential
2643:) = 1 on
2078:equilibrium measure
2039:{\displaystyle n=2}
1731:{\displaystyle n=2}
1680:Newtonian potential
1450:in the definition:
1237:tends to infinity:
930:Newtonian potential
920:) â 0 as
881:) = 0 on
372:boundary conditions
2684:indicator function
2620:is defined as the
2592:
2501:
2426:
2324:
2295:
2256:
2233:
2206:
2149:
2074:capacitary measure
2062:
2036:
2007:
1932:
1903:
1810:
1728:
1698:
1676:capacity potential
1656:conformal capacity
1644:
1607:
1440:
1400:
1306:
1277:
1189:
1140:
1040:
938:Newtonian capacity
841:
737:Dirichlet's energy
706:
576:
501:
338:
200:condenser capacity
138:
96:Condenser capacity
69:condenser capacity
65:Newtonian capacity
2804:, available from
2700:Analytic capacity
2446:energy functional
2197:
2005:
1901:
1564:
1532:
1497:
1323:of the conductor
1262:
1126:
934:harmonic capacity
889:Harmonic capacity
798:
660:
519:normal derivative
457:
360:harmonic function
319:
284:
214:) or cap(ÎŁ,
202:of ÎŁ relative to
49:electrical charge
33:capacity of a set
18:Energy functional
16:(Redirected from
3012:
3005:Potential theory
2991:
2973:
2961:"Robin constant"
2955:
2937:
2914:
2898:
2855:
2813:Doob, Joseph Leo
2803:
2771:Choquet, Gustave
2765:
2748:
2721:Potential theory
2705:
2676:obstacle problem
2670:with respect to
2601:
2599:
2598:
2593:
2588:
2567:
2566:
2548:
2547:
2510:
2508:
2507:
2502:
2435:
2433:
2432:
2427:
2377:
2376:
2333:
2331:
2330:
2325:
2304:
2302:
2301:
2296:
2266:concentrated on
2265:
2263:
2262:
2257:
2242:
2240:
2239:
2234:
2232:
2231:
2223:
2219:
2205:
2164:Dirichlet energy
2158:
2156:
2155:
2150:
2118:
2117:
2071:
2069:
2068:
2063:
2045:
2043:
2042:
2037:
2016:
2014:
2013:
2008:
2006:
2004:
2003:
1989:
1980:
1941:
1939:
1938:
1933:
1912:
1910:
1909:
1904:
1902:
1900:
1899:
1898:
1887:
1872:
1863:
1819:
1817:
1816:
1811:
1776:
1775:
1740:Green's function
1737:
1735:
1734:
1729:
1707:
1705:
1704:
1699:
1660:conformal radius
1653:
1651:
1650:
1645:
1616:
1614:
1613:
1608:
1603:
1598:
1597:
1592:
1580:
1575:
1574:
1565:
1563:
1552:
1547:
1539:
1533:
1531:
1523:
1515:
1513:
1512:
1511:
1498:
1496:
1485:
1449:
1447:
1446:
1441:
1409:
1407:
1406:
1401:
1399:
1398:
1315:
1313:
1312:
1307:
1299:
1298:
1276:
1198:
1196:
1195:
1190:
1188:
1187:
1182:
1149:
1147:
1146:
1141:
1133:
1127:
1125:
1117:
1109:
1107:
1106:
1049:
1047:
1046:
1041:
1033:
1028:
1027:
1022:
1010:
1005:
1004:
997:
996:
991:
850:
848:
847:
842:
837:
832:
831:
826:
814:
809:
808:
799:
797:
796:
795:
767:
715:
713:
712:
707:
699:
694:
693:
688:
676:
671:
670:
661:
659:
658:
657:
629:
585:
583:
582:
577:
575:
574:
569:
530:
510:
508:
507:
502:
458:
456:
448:
440:
424:
400:
347:
345:
344:
339:
334:
326:
320:
318:
310:
302:
300:
299:
298:
285:
283:
282:
281:
253:
194:) is known as a
188:electromagnetism
158:will denote the
157:
147:
145:
144:
139:
137:
136:
131:
57:potential energy
41:Lebesgue measure
21:
3020:
3019:
3015:
3014:
3013:
3011:
3010:
3009:
2995:
2994:
2976:
2958:
2940:
2927:
2906:
2899:, available at
2864:Stampacchia, G.
2858:
2837:
2811:
2769:
2746:
2739:
2736:
2703:
2696:
2558:
2539:
2519:
2518:
2469:
2468:
2458:
2442:
2440:Generalizations
2362:
2339:
2338:
2307:
2306:
2272:
2271:
2248:
2247:
2196:
2192:
2191:
2171:
2170:
2109:
2089:
2088:
2054:
2053:
2022:
2021:
1984:
1947:
1946:
1918:
1917:
1882:
1867:
1836:
1835:
1767:
1747:
1746:
1714:
1713:
1684:
1683:
1668:
1630:
1629:
1587:
1566:
1556:
1540:
1524:
1516:
1504:
1499:
1489:
1455:
1454:
1420:
1419:
1416:
1378:
1358:
1357:
1340:Wiener capacity
1290:
1242:
1241:
1228:
1215:
1177:
1172:
1171:
1161:
1118:
1110:
1098:
1078:
1077:
1017:
986:
981:
961:
960:
891:
821:
800:
787:
771:
746:
745:
683:
662:
649:
633:
602:
601:
564:
559:
558:
549: â Î(
544:
528:
449:
441:
434:
433:
422:
412:is the outward
398:
327:
311:
303:
291:
286:
273:
257:
223:
222:
149:
126:
121:
120:
98:
93:
81:Gustave Choquet
77:
75:Historical note
37:Euclidean space
23:
22:
15:
12:
11:
5:
3018:
3016:
3008:
3007:
2997:
2996:
2993:
2992:
2974:
2956:
2938:
2925:
2904:
2868:Weinberger, H.
2856:
2835:
2809:
2787:(4): 385â397,
2767:
2735:
2732:
2731:
2730:
2727:Choquet theory
2724:
2718:
2712:
2706:
2695:
2692:
2603:
2602:
2591:
2587:
2582:
2579:
2576:
2573:
2570:
2565:
2561:
2557:
2554:
2551:
2546:
2542:
2538:
2535:
2532:
2529:
2526:
2512:
2511:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2457:
2454:
2441:
2438:
2437:
2436:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2375:
2372:
2369:
2365:
2361:
2358:
2355:
2352:
2349:
2346:
2323:
2320:
2317:
2314:
2294:
2291:
2288:
2285:
2282:
2279:
2255:
2244:
2243:
2230:
2227:
2222:
2218:
2215:
2212:
2209:
2204:
2200:
2195:
2190:
2187:
2184:
2181:
2178:
2160:
2159:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2116:
2112:
2108:
2105:
2102:
2099:
2096:
2072:is called the
2061:
2035:
2032:
2029:
2018:
2017:
2002:
1998:
1995:
1992:
1988:
1983:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1931:
1928:
1925:
1914:
1913:
1897:
1894:
1891:
1886:
1881:
1878:
1875:
1871:
1866:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1821:
1820:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1774:
1770:
1766:
1763:
1760:
1757:
1754:
1727:
1724:
1721:
1697:
1694:
1691:
1674:is called the
1667:
1664:
1643:
1640:
1637:
1618:
1617:
1606:
1602:
1596:
1591:
1586:
1583:
1579:
1573:
1569:
1562:
1559:
1555:
1550:
1546:
1543:
1538:
1530:
1527:
1522:
1519:
1510:
1507:
1502:
1495:
1492:
1488:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1439:
1436:
1433:
1430:
1427:
1415:
1412:
1411:
1410:
1397:
1394:
1391:
1388:
1385:
1381:
1377:
1374:
1371:
1368:
1365:
1344:Robin constant
1317:
1316:
1305:
1302:
1297:
1293:
1289:
1286:
1283:
1280:
1275:
1272:
1269:
1265:
1261:
1258:
1255:
1252:
1249:
1224:
1211:
1186:
1181:
1157:
1151:
1150:
1139:
1136:
1132:
1124:
1121:
1116:
1113:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1051:
1050:
1039:
1036:
1032:
1026:
1021:
1016:
1013:
1009:
1003:
1000:
995:
990:
984:
980:
977:
974:
971:
968:
890:
887:
852:
851:
840:
836:
830:
825:
820:
817:
813:
807:
803:
794:
790:
786:
783:
780:
777:
774:
770:
765:
762:
759:
756:
753:
717:
716:
705:
702:
698:
692:
687:
682:
679:
675:
669:
665:
656:
652:
648:
645:
642:
639:
636:
632:
627:
624:
621:
618:
615:
612:
609:
588:
587:
573:
568:
545: = 2
540:
534:
533:
514:
513:
512:
511:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
455:
452:
447:
444:
428:
427:
407:
394:
366:between ÎŁ and
358:is the unique
349:
348:
337:
333:
330:
325:
317:
314:
309:
306:
297:
294:
289:
280:
276:
272:
269:
266:
263:
260:
256:
251:
248:
245:
242:
239:
236:
233:
230:
135:
130:
97:
94:
92:
89:
76:
73:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3017:
3006:
3003:
3002:
3000:
2990:
2986:
2985:
2980:
2975:
2972:
2968:
2967:
2962:
2957:
2954:
2950:
2949:
2944:
2939:
2936:
2932:
2928:
2926:0-521-46654-7
2922:
2918:
2913:
2912:
2905:
2902:
2897:
2893:
2889:
2885:
2882:(12): 43â77,
2881:
2878:, Serie III,
2877:
2873:
2869:
2865:
2861:
2857:
2854:
2850:
2846:
2842:
2838:
2836:0-387-90881-1
2832:
2828:
2824:
2820:
2819:
2814:
2810:
2807:
2802:
2798:
2794:
2790:
2786:
2783:(in French),
2782:
2781:
2776:
2772:
2768:
2764:
2760:
2756:
2752:
2745:
2744:
2738:
2737:
2733:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2701:
2698:
2697:
2693:
2691:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2660:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2627:
2623:
2619:
2615:
2611:
2606:
2589:
2577:
2568:
2563:
2555:
2544:
2540:
2536:
2530:
2524:
2517:
2516:
2515:
2498:
2495:
2489:
2483:
2477:
2467:
2466:
2465:
2463:
2455:
2453:
2451:
2447:
2439:
2420:
2414:
2411:
2405:
2399:
2396:
2390:
2387:
2384:
2378:
2373:
2370:
2367:
2363:
2359:
2356:
2350:
2344:
2337:
2336:
2335:
2318:
2312:
2292:
2289:
2283:
2277:
2269:
2253:
2228:
2225:
2220:
2213:
2207:
2202:
2193:
2188:
2182:
2176:
2169:
2168:
2167:
2165:
2143:
2137:
2134:
2128:
2122:
2119:
2114:
2110:
2106:
2100:
2094:
2087:
2086:
2085:
2083:
2082:Borel measure
2079:
2075:
2059:
2052:
2047:
2033:
2030:
2027:
1996:
1993:
1990:
1981:
1976:
1973:
1970:
1964:
1961:
1958:
1952:
1945:
1944:
1943:
1929:
1926:
1923:
1895:
1892:
1889:
1879:
1876:
1873:
1864:
1859:
1853:
1850:
1847:
1841:
1834:
1833:
1832:
1830:
1826:
1804:
1798:
1795:
1789:
1786:
1783:
1777:
1772:
1768:
1764:
1758:
1752:
1745:
1744:
1743:
1741:
1725:
1722:
1719:
1711:
1695:
1692:
1689:
1681:
1677:
1673:
1665:
1663:
1661:
1657:
1641:
1635:
1627:
1623:
1604:
1594:
1584:
1571:
1567:
1560:
1557:
1553:
1548:
1544:
1541:
1528:
1520:
1508:
1505:
1500:
1493:
1490:
1486:
1481:
1478:
1472:
1469:
1460:
1453:
1452:
1451:
1434:
1431:
1428:
1413:
1392:
1386:
1383:
1379:
1375:
1369:
1363:
1356:
1355:
1354:
1352:
1348:
1345:
1341:
1336:
1334:
1330:
1326:
1322:
1303:
1295:
1291:
1287:
1278:
1267:
1259:
1253:
1247:
1240:
1239:
1238:
1236:
1232:
1227:
1223:
1220:and (ÎŁ,
1219:
1216:will enclose
1214:
1210:
1206:
1202:
1184:
1169:
1165:
1160:
1156:
1137:
1134:
1122:
1114:
1103:
1099:
1095:
1089:
1083:
1076:
1075:
1074:
1072:
1068:
1064:
1060:
1056:
1037:
1034:
1024:
1014:
1001:
993:
982:
978:
972:
966:
959:
958:
957:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
903:
899:
895:
894:Heuristically
888:
886:
884:
880:
876:
872:
868:
864:
860:
857:
838:
828:
818:
805:
801:
792:
788:
781:
778:
775:
768:
763:
757:
751:
744:
743:
742:
741:
738:
734:
730:
726:
722:
703:
700:
690:
680:
667:
663:
654:
650:
643:
640:
637:
630:
625:
619:
616:
607:
600:
599:
598:
596:
592:
571:
556:
552:
548:
543:
539:
536:
535:
531:
524:
520:
516:
515:
495:
489:
486:
480:
474:
468:
462:
453:
445:
432:
431:
430:
429:
425:
418:
415:
411:
408:
405:
401:
395:
392:
388:
384:
380:
376:
373:
369:
365:
361:
357:
354:
353:
352:
335:
331:
328:
315:
307:
295:
292:
287:
278:
274:
267:
264:
261:
254:
249:
246:
240:
237:
228:
221:
220:
219:
217:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
162:-dimensional
161:
156:
152:
133:
118:
114:
111:
107:
103:
95:
90:
88:
86:
82:
74:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
2982:
2964:
2946:
2910:
2879:
2875:
2817:
2784:
2778:
2742:
2687:
2679:
2671:
2667:
2663:
2661:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2617:
2613:
2609:
2607:
2604:
2513:
2459:
2443:
2267:
2245:
2161:
2077:
2073:
2048:
2019:
1915:
1828:
1824:
1822:
1709:
1675:
1671:
1669:
1655:
1625:
1621:
1619:
1417:
1353:is given by
1350:
1346:
1343:
1339:
1337:
1332:
1328:
1324:
1318:
1234:
1225:
1221:
1217:
1212:
1208:
1204:
1200:
1167:
1158:
1154:
1152:
1070:
1066:
1062:
1054:
1052:
953:
949:
945:
941:
937:
933:
925:
921:
917:
913:
909:
905:
901:
897:
892:
882:
878:
874:
870:
866:
862:
858:
853:
728:
724:
718:
594:
590:
589:
550:
546:
541:
537:
526:
522:
420:
409:
403:
396:
390:
386:
382:
378:
374:
367:
363:
355:
350:
215:
211:
207:
203:
199:
191:
183:
182:be another (
179:
159:
154:
150:
116:
113:hypersurface
105:
99:
85:Choquet 1986
78:
68:
64:
60:
32:
26:
2860:Littman, W.
2709:Capacitance
2616:containing
1626:logarithmic
1624:, the term
1162:denote the
908:satisfying
555:unit sphere
414:unit normal
110:dimensional
104:, smooth, (
100:Let ÎŁ be a
91:Definitions
53:capacitance
29:mathematics
2943:"Capacity"
2935:0828.31001
2896:0116.30302
2853:0549.31001
2823:Heidelberg
2801:0607.01017
2763:0257.31001
2734:References
1666:Properties
1166:of radius
944:, denoted
740:functional
206:, denoted
2989:EMS Press
2971:EMS Press
2953:EMS Press
2575:∇
2553:∇
2541:∫
2487:∇
2478:⋅
2475:∇
2415:λ
2400:λ
2388:−
2371:×
2364:∫
2360:∫
2351:λ
2319:λ
2305:and with
2278:λ
2254:λ
2226:−
2214:λ
2203:λ
2138:μ
2123:μ
2111:∫
2060:μ
1994:−
1977:
1962:−
1927:≥
1893:−
1877:−
1851:−
1799:μ
1787:−
1769:∫
1693:≥
1639:→
1582:∇
1568:∫
1561:π
1542:σ
1529:ν
1526:∂
1518:∂
1501:∫
1494:π
1482:−
1467:Σ
1432:−
1384:−
1285:Σ
1274:∞
1271:→
1199:. Since
1135:σ
1123:ν
1120:∂
1112:∂
1100:∫
1012:∇
999:∖
983:∫
952:) or cap(
854:over all
816:∇
802:∫
789:σ
779:−
731:) is the
727:(ÎŁ,
678:∇
664:∫
651:σ
641:−
614:Σ
593:(ÎŁ,
490:ν
487:⋅
472:∇
454:ν
451:∂
443:∂
389:) = 0 on
370:with the
329:σ
316:ν
313:∂
305:∂
288:∫
275:σ
265:−
250:−
235:Σ
210:(ÎŁ,
196:condenser
2999:Category
2870:(1963),
2827:xxiv+846
2815:(1984),
2773:(1986),
2694:See also
1708:and the
1545:′
1509:′
332:′
296:′
176:boundary
61:harmonic
2888:0161019
2845:0731258
2806:Gallica
2793:0867115
2755:0259146
2622:infimum
2051:measure
1831:, and
928:is the
735:of the
733:infimum
525:across
517:is the
351:where:
198:. The
178:. Let
172:bounded
166:(i.e.,
164:compact
2933:
2923:
2901:NUMDAM
2894:
2886:
2851:
2843:
2833:
2799:
2791:
2761:
2753:
2647:; and
1678:, the
1164:sphere
168:closed
102:closed
45:volume
31:, the
2747:(PDF)
2635:with
1916:when
1823:with
1712:when
1682:when
1231:limit
1057:is a
865:with
532:; and
417:field
153:â„ 3;
2921:ISBN
2831:ISBN
2049:The
2020:for
1942:and
1347:W(K)
1338:The
170:and
2931:Zbl
2892:Zbl
2849:Zbl
2797:Zbl
2759:Zbl
2686:of
2678:on
2666:of
2631:on
2199:inf
2076:or
1974:log
1742:as
1349:of
1342:or
1264:lim
1233:as
1053:If
940:of
936:or
861:on
557:in
521:of
426:and
419:to
115:in
87:).
63:or
35:in
27:In
3001::
2987:,
2981:,
2969:,
2963:,
2951:,
2945:,
2929:,
2919:,
2890:,
2884:MR
2880:17
2874:,
2866:;
2862:;
2847:,
2841:MR
2839:,
2829:,
2795:,
2789:MR
2777:,
2757:,
2751:MR
2659:.
2452:.
2046:.
1662:.
1207:,
1073::
885:.
723::
148:,
71:.
2903:.
2785:3
2688:E
2680:D
2672:D
2668:E
2657:D
2653:x
2651:(
2649:v
2645:E
2641:x
2639:(
2637:v
2633:D
2629:v
2618:E
2614:D
2610:E
2590:x
2586:d
2581:)
2578:u
2572:(
2569:A
2564:T
2560:)
2556:u
2550:(
2545:D
2537:=
2534:]
2531:u
2528:[
2525:I
2499:0
2496:=
2493:)
2490:u
2484:A
2481:(
2424:)
2421:y
2418:(
2412:d
2409:)
2406:x
2403:(
2397:d
2394:)
2391:y
2385:x
2382:(
2379:G
2374:K
2368:K
2357:=
2354:)
2348:(
2345:E
2322:)
2316:(
2313:E
2293:1
2290:=
2287:)
2284:K
2281:(
2268:K
2229:1
2221:]
2217:)
2211:(
2208:E
2194:[
2189:=
2186:)
2183:K
2180:(
2177:C
2147:)
2144:S
2141:(
2135:=
2132:)
2129:y
2126:(
2120:d
2115:S
2107:=
2104:)
2101:K
2098:(
2095:C
2034:2
2031:=
2028:n
2001:|
1997:y
1991:x
1987:|
1982:1
1971:=
1968:)
1965:y
1959:x
1956:(
1953:G
1930:3
1924:n
1896:2
1890:n
1885:|
1880:y
1874:x
1870:|
1865:1
1860:=
1857:)
1854:y
1848:x
1845:(
1842:G
1829:S
1825:x
1808:)
1805:y
1802:(
1796:d
1793:)
1790:y
1784:x
1781:(
1778:G
1773:S
1765:=
1762:)
1759:x
1756:(
1753:u
1726:2
1723:=
1720:n
1696:3
1690:n
1672:u
1642:2
1636:n
1605:x
1601:d
1595:2
1590:|
1585:u
1578:|
1572:D
1558:2
1554:1
1549:=
1537:d
1521:u
1506:S
1491:2
1487:1
1479:=
1476:)
1473:S
1470:,
1464:(
1461:C
1438:)
1435:2
1429:n
1426:(
1396:)
1393:K
1390:(
1387:W
1380:e
1376:=
1373:)
1370:K
1367:(
1364:C
1351:K
1333:K
1331:(
1329:C
1325:K
1304:.
1301:)
1296:r
1292:S
1288:,
1282:(
1279:C
1268:r
1260:=
1257:)
1254:K
1251:(
1248:C
1235:r
1226:r
1222:S
1218:K
1213:r
1209:S
1205:r
1201:K
1185:n
1180:R
1168:r
1159:r
1155:S
1138:.
1131:d
1115:u
1104:S
1096:=
1093:)
1090:K
1087:(
1084:C
1071:u
1067:S
1063:K
1055:S
1038:.
1035:x
1031:d
1025:2
1020:|
1015:u
1008:|
1002:K
994:n
989:R
979:=
976:)
973:K
970:(
967:C
954:K
950:K
948:(
946:C
942:K
926:u
922:x
918:x
916:(
914:u
910:u
906:K
902:u
898:K
883:S
879:x
877:(
875:v
871:x
869:(
867:v
863:D
859:v
839:x
835:d
829:2
824:|
819:v
812:|
806:D
793:n
785:)
782:2
776:n
773:(
769:1
764:=
761:]
758:v
755:[
752:I
729:S
725:C
704:.
701:x
697:d
691:2
686:|
681:u
674:|
668:D
655:n
647:)
644:2
638:n
635:(
631:1
626:=
623:)
620:S
617:,
611:(
608:C
595:S
591:C
586:.
572:n
567:R
551:n
547:Ï
542:n
538:Ï
529:âČ
527:S
523:u
499:)
496:x
493:(
484:)
481:x
478:(
475:u
469:=
466:)
463:x
460:(
446:u
423:âČ
421:S
410:Μ
406:;
404:S
399:âČ
397:S
393:;
391:S
387:x
385:(
383:u
379:x
377:(
375:u
368:S
364:D
356:u
336:,
324:d
308:u
293:S
279:n
271:)
268:2
262:n
259:(
255:1
247:=
244:)
241:S
238:,
232:(
229:C
216:S
212:S
208:C
204:S
192:S
184:n
180:S
160:n
155:K
151:n
134:n
129:R
117:n
106:n
20:)
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