Convexity estimates for the Green’s function
In this paper, for the Green’s function of a bounded convex domain Ω with pole at x 0 ∈ Ω , we find the auxiliary curvature function which satisfies a differential inequality and so attains its minimum on the boundary. Moreover, we obtain a convexity estimate for the Green’s function of the domain a...
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| Published in | Calculus of variations and partial differential equations Vol. 53; no. 3-4; pp. 675 - 688 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2015
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | In this paper, for the Green’s function of a bounded convex domain
Ω
with pole at
x
0
∈
Ω
, we find the auxiliary curvature function which satisfies a differential inequality and so attains its minimum on the boundary. Moreover, we obtain a convexity estimate for the Green’s function of the domain above and give the proof of its specific convexity from the viewpoint of partial differential equations themselves. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) In this paper, for the Green's function of a bounded convex domain ... with pole at ..., we find the auxiliary curvature function which satisfies a differential inequality and so attains its minimum on the boundary. Moreover, we obtain a convexity estimate for the Green's function of the domain above and give the proof of its specific convexity from the viewpoint of partial differential equations themselves. In this paper, for the Green’s function of a bounded convex domain Ω with pole at x 0 ∈ Ω , we find the auxiliary curvature function which satisfies a differential inequality and so attains its minimum on the boundary. Moreover, we obtain a convexity estimate for the Green’s function of the domain above and give the proof of its specific convexity from the viewpoint of partial differential equations themselves. |
| Author | Shi, Shujun |
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| Cites_doi | 10.1002/cpa.20197 10.1090/S0002-9947-2012-05436-5 10.1214/aop/1176989412 10.1007/BF01405053 10.1016/0022-1236(76)90004-5 10.1512/iumj.2011.60.4222 10.1080/03605302.2012.727129 10.2307/2371223 10.1002/cpa.20318 10.1007/BF00946979 10.1112/jlms/s1-32.3.286 10.24033/asens.1480 10.1090/conm/367/06751 10.1007/BFb0075060 |
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| References | Ma, Shi, Ye (CR14) 2012; 37 Ma, Ou, Zhang (CR13) 2010; 63 CR2 Brascamp, Lieb (CR6) 1976; 22 Acker, Payne, Philippin (CR1) 1981; 32 Borell (CR5) 1993; 21 Makar-Limanov (CR15) 1971; 9 Caffarelli, Guan, Ma (CR7) 2007; 60 Borell (CR4) 1984; 17 CR12 Gabriel (CR8) 1957; 32 CR10 Jost, Ma, Ou (CR11) 2012; 364 Gergen (CR9) 1931; 53 Bian, Guan, Ma, Xu (CR3) 2011; 60 Trudinger (CR16) 1997; 488 L Caffarelli (763_CR7) 2007; 60 XN Ma (763_CR14) 2012; 37 J Gergen (763_CR9) 1931; 53 XN Ma (763_CR13) 2010; 63 A Acker (763_CR1) 1981; 32 R Gabriel (763_CR8) 1957; 32 763_CR12 763_CR10 NS Trudinger (763_CR16) 1997; 488 BJ Bian (763_CR3) 2011; 60 763_CR2 C Borell (763_CR4) 1984; 17 J Jost (763_CR11) 2012; 364 LG Makar-Limanov (763_CR15) 1971; 9 C Borell (763_CR5) 1993; 21 HJ Brascamp (763_CR6) 1976; 22 |
| References_xml | – volume: 488 start-page: 203 year: 1997 end-page: 220 ident: CR16 article-title: On new isoperimetric inequalities and symmetrization publication-title: J. Reine Angew. Math. contributor: fullname: Trudinger – volume: 60 start-page: 1769 year: 2007 end-page: 1791 ident: CR7 article-title: A constant rank theorem for solutions of fully nonlinear elliptic equations publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20197 contributor: fullname: Ma – volume: 17 start-page: 451 year: 1984 end-page: 467 ident: CR4 article-title: Hitting probabilities of killed Brownian motion: a study on geometric regularit publication-title: Ann. Sci. Ecole Norm. Sup. contributor: fullname: Borell – volume: 364 start-page: 4605 year: 2012 end-page: 4627 ident: CR11 article-title: Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings publication-title: Trans. AMS doi: 10.1090/S0002-9947-2012-05436-5 contributor: fullname: Ou – volume: 21 start-page: 482 year: 1993 end-page: 489 ident: CR5 article-title: Geometric properties of some familiar diffusions in publication-title: Ann. Probab. doi: 10.1214/aop/1176989412 contributor: fullname: Borell – volume: 9 start-page: 52 year: 1971 end-page: 53 ident: CR15 article-title: Solution of Dirichlet’s problem for the equation on a convex region publication-title: Math. Notes Acad. Sci. USSR doi: 10.1007/BF01405053 contributor: fullname: Makar-Limanov – volume: 22 start-page: 366 year: 1976 end-page: 389 ident: CR6 article-title: On extensions of the Brunn–Minkowski and Prekopa–Leindler theorems, including inequalities for log-concave functions, with an application to the diffusion equation publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(76)90004-5 contributor: fullname: Lieb – ident: CR2 – volume: 60 start-page: 101 year: 2011 end-page: 120 ident: CR3 article-title: A constant rank theorem for quasiconcave solutions of fully nonlinear partial differential equations publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2011.60.4222 contributor: fullname: Xu – ident: CR12 – ident: CR10 – volume: 37 start-page: 2116 year: 2012 end-page: 2157 ident: CR14 article-title: The convexity estimates for the solutions of two elliptic equations publication-title: Commun. Partial Diff. Equ. doi: 10.1080/03605302.2012.727129 contributor: fullname: Ye – volume: 53 start-page: 746 year: 1931 end-page: 752 ident: CR9 article-title: Note on the Green function of a star-shaped three dimensional region publication-title: Am. J. 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Anal. doi: 10.1016/0022-1236(76)90004-5 contributor: fullname: HJ Brascamp – volume: 488 start-page: 203 year: 1997 ident: 763_CR16 publication-title: J. Reine Angew. Math. contributor: fullname: NS Trudinger – volume: 60 start-page: 1769 year: 2007 ident: 763_CR7 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20197 contributor: fullname: L Caffarelli – volume: 9 start-page: 52 year: 1971 ident: 763_CR15 publication-title: Math. Notes Acad. Sci. USSR doi: 10.1007/BF01405053 contributor: fullname: LG Makar-Limanov – ident: 763_CR2 – volume: 53 start-page: 746 year: 1931 ident: 763_CR9 publication-title: Am. J. Math. doi: 10.2307/2371223 contributor: fullname: J Gergen – volume: 63 start-page: 935 year: 2010 ident: 763_CR13 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20318 contributor: fullname: XN Ma – volume: 32 start-page: 286 year: 1957 ident: 763_CR8 publication-title: J. London Math. 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Probab. doi: 10.1214/aop/1176989412 contributor: fullname: C Borell – ident: 763_CR12 doi: 10.1007/BFb0075060 – volume: 60 start-page: 101 year: 2011 ident: 763_CR3 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2011.60.4222 contributor: fullname: BJ Bian |
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Ω
with pole at
x
0
∈
Ω
, we find the auxiliary curvature function which satisfies a... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) In this paper, for the Green's function of a bounded convex domain ... with pole at... |
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| SubjectTerms | Analysis Calculus of variations Calculus of Variations and Optimal Control; Optimization Control Mathematical and Computational Physics Mathematics Mathematics and Statistics Partial differential equations Systems Theory Theoretical |
| Title | Convexity estimates for the Green’s function |
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