Polyharmonic capacity and Wiener test of higher order
In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmon...
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| Vydáno v: | Inventiones mathematicae Ročník 211; číslo 2; s. 779 - 853 |
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| Abstract | In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Maz’ya (Invent Math 175(2):287–334,
2009
). As a preliminary stage of this work, in Mayboroda and Maz’ya (Invent Math 196(1):168,
2014
) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order. |
|---|---|
| AbstractList | In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Maz’ya (Invent Math 175(2):287–334,
2009
). As a preliminary stage of this work, in Mayboroda and Maz’ya (Invent Math 196(1):168,
2014
) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order. In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Mazya (Invent Math 175(2):287-334, 2009). As a preliminary stage of this work, in Mayboroda and Mazya (Invent Math 196(1):168, 2014) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order. |
| Author | Mayboroda, Svitlana Maz’ya, Vladimir |
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Helv.1993683385414123676110.1007/BF025658270794.31005 Maz’jaVThe continuity at a boundary point of the solutions of quasi-linear elliptic equationsVestn. Leningr. Univ.197025134255274948(Russian) MayborodaSMaz’yaVBoundedness of the gradient of a solution and Wiener test of order one for the biharmonic equationInvent. Math.20091752287334247010910.1007/s00222-008-0150-x1166.35014 PipherJVerchotaGMaximum principles for the polyharmonic equation on Lipschitz domainsPotential Anal.199546615636136138010.1007/BF023458280844.35013 MirandaCTeorema del massimo modulo e teorema di esistenza e di unicità per il problema di Dirichlet relativo alle equazioni ellittiche in due variabiliAnn. Mat. Pura Appl. (4)19584626531112461510.1007/BF024129190093.29403 Mayboroda, S., Maz’ya, V.: Pointwise estimates for the polyharmonic Green function in general domains. In: Cialdea, A., (ed.) et al. Analysis, Partial Differential Equations and Applications. The Vladimir Maz’ya anniversary volume. 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Soc.196066778012461810.1090/S0002-9904-1960-10402-80091.27301 KozlovVMaz’yaVRossmannJSpectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations2001ProvidenceAmerican Mathematical Society0965.35003 AntmanSSNonlinear Problems of Elasticity20052New YorkSpringer1098.74001 Maz’yaVRossmannJOn the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^n$$\end{document} with conical pointsAnn. Glob. Anal. Geom.1992102125150117591510.1007/BF001309160794.35043 MalýJZiemerWPFine Regularity of Solutions of Elliptic Partial Differential Equations1997ProvidenceAmerican Mathematical Society0882.35001 LabutinDPotential estimates for a class of fully nonlinear elliptic equationsDuke Math. 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Fourier (Grenoble)198636310913586566310.5802/aif.10620589.35040 FabesEJerisonDKenigCThe Wiener test for degenerate elliptic equationsAnn. Inst. Fourier (Grenoble)198232315118268802410.5802/aif.8830488.35034 MeleshkoVVSelected topics in the history of the two-dimensional biharmonic problemAppl. Mech. Rev.200356338510.1115/1.1521166 ShenZNecessary and sufficient conditions for the solvability of the Lp Dirichlet problem on Lipschitz domainsMath. Ann.20063363697725224976510.1007/s00208-006-0022-x1194.35131 Maz’ya, V., Nazarov, S. A., Plamenevskiĭ, B. A.: Singularities of solutions of the Dirichlet problem in the exterior of a thin cone, Mat. Sb. (N.S.) 122 (164), vol. 4, pp. 435–457 (1983) English translation: Math. USSR-Sb. 50 (1985), no. 2, 415–437 Dal MasoGMoscoUWiener criteria and energy decay for relaxed Dirichlet problemsArch. Ration. Mech. Anal.198695434538785378310.1007/BF002768410634.35033 PipherJVerchotaGThe Dirichlet problem in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} for the biharmonic equation on Lipschitz domainsAm. J. Math.19921145923972118352710.2307/23748850768.31006 CiarletPhMathematical Elasticity. Vol II. Theory of Plates1997AmsterdamNorth-Holland Publishing Co.0888.73001 MayborodaSMaz’yaVRegularity of solutions to the polyharmonic equation in general domainsInvent. Math.20141961168317957210.1007/s00222-013-0464-11292.35114 AdamsDRLp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} potential theory techniques and nonlinear PDE, Potential theory (Nagoya, 1990), 1–151992Berlinde Gruyter PoincaréHSur les équations aux derivées partielle de la physique mathématiqueAm. J. Math.18901221129910.2307/236962022.0977.03 ShenZOn estimates of biharmonic functions on Lipschitz and convex domainsJ. Geom. Anal.2006164721734227195110.1007/BF029221381174.31306 H Poincaré (756_CR40) 1890; 12 SC Zaremba (756_CR47) 1909 S Mayboroda (756_CR24) 2014; 196 J Pipher (756_CR38) 1993; 68 V Maz’ja (756_CR25) 1970; 25 S Agmon (756_CR5) 1959; 12 SS Antman (756_CR6) 2005 C Miranda (756_CR35) 1958; 46 J Pipher (756_CR37) 1992; 114 756_CR8 W Littman (756_CR20) 1963; 17 T Kilpeläinen (756_CR15) 1994; 172 VV Meleshko (756_CR33) 2003; 56 G Maso Dal (756_CR11) 1986; 95 N Trudinger (756_CR45) 2002; 124 Z Shen (756_CR43) 2006; 336 B Dahlberg (756_CR10) 1986; 36 756_CR18 756_CR30 P Lindqvist (756_CR19) 1985; 155 E Fabes (756_CR14) 1982; 32 V Maz’ya (756_CR29) 2002; 115 E Fabes (756_CR13) 1989; 59 S Mayboroda (756_CR22) 2009; 175 V Maz’ya (756_CR32) 1992; 10 S Agmon (756_CR4) 1960; 66 D Labutin (756_CR17) 2002; 111 J Malý (756_CR21) 1997 S-YA Chang (756_CR7) 2005; 42 V Kozlov (756_CR16) 2001 D Adams (756_CR3) 1996 DR Adams (756_CR1) 1992 756_CR2 LC Evans (756_CR12) 1982; 78 756_CR27 J Nečas (756_CR36) 1967 756_CR26 J Pipher (756_CR39) 1995; 4 C Miranda (756_CR34) 1948; 2 756_CR28 756_CR41 N Wiener (756_CR46) 1924; 3 756_CR23 756_CR44 Ph Ciarlet (756_CR9) 1997 V Maz’ya (756_CR31) 1991; 9 Z Shen (756_CR42) 2006; 16 |
| References_xml | – reference: WienerNThe Dirichlet problemJ. Math. Phys.1924312714610.1002/sapm19243312751.0361.01 – reference: EvansLCGariepyRFWiener’s criterion for the heat equationArch. Ration. Mech. Anal.198278429331465354410.1007/BF002495830508.35038 – reference: PipherJVerchotaGA maximum principle for biharmonic functions in Lipschitz and C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} domainsComment. Math. Helv.1993683385414123676110.1007/BF025658270794.31005 – reference: AdamsDHedbergLFunction Spaces and Potential Theory1996BerlinSpringer0834.46021 – reference: MirandaCFormule di maggiorazione e teorema di esistenza per le funzioni biarmoniche de due variabiliGiorn. Mat. Battagl. (4)194827897118300580037.07103 – reference: ShenZNecessary and sufficient conditions for the solvability of the Lp Dirichlet problem on Lipschitz domainsMath. Ann.20063363697725224976510.1007/s00208-006-0022-x1194.35131 – reference: Maz’yaVRossmannJOn the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domainsAnn. Glob. Anal. Geom.199193253303114340610.1007/BF001368150753.35013 – reference: ZarembaSCSur le principe du minimum1909CracovieBull. Acad. Sci40.0451.01 – reference: MayborodaSMaz’yaVBoundedness of the gradient of a solution and Wiener test of order one for the biharmonic equationInvent. Math.20091752287334247010910.1007/s00222-008-0150-x1166.35014 – reference: Maz’ya, V., Nazarov, S. A., Plamenevskiĭ, B. A.: Singularities of solutions of the Dirichlet problem in the exterior of a thin cone, Mat. Sb. (N.S.) 122 (164), vol. 4, pp. 435–457 (1983) English translation: Math. 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