Geodesic
Isoperimetric properties of Euclidean boundary moments of a~simply connected domain
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2013), pp. 66-79

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We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For $L^p$-norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.
Keywords: distance function to the boundary of a domain, Bonnesen inequality, isoperimetric inequalities, Euclidean moments of a domain with respect to the boundary, torsional rigidity, isoperimetric monotonicity. 
@article{IVM_2013_8_a6,
     author = {R. G. Salakhudinov},
     title = {Isoperimetric properties of {Euclidean} boundary moments of a~simply connected domain},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {66--79},
     publisher = {mathdoc},
     number = {8},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_8_a6/}
}
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R. G. Salakhudinov. Isoperimetric properties of Euclidean boundary moments of a~simply connected domain. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2013), pp. 66-79. http://geodesic.mathdoc.fr/item/IVM_2013_8_a6/