Page 381 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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SPECTRAL THEORY AND INTEGRABLE SYSTEMS (MS-57)
Computing Eigenvalues of the Laplacian on Rough Domains
Frank Rösler, roslerf@cardiff.ac.uk
Cardiff University, United Kingdom
Coauthor: Alexei Stepanenko
We discuss a recent work in which we prove a general Mosco convergence theorem for bounded
Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this
notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures
spectral convergence. A key element of the proof is the development of a novel, explicit
Poincaré-type inequality, which is of independent interest.
These results are applied construct a universal algorithm capable of computing the eigenval-
ues of the Dirichlet Laplacian on a wide class of rough domains. This immediately leads to new
classifications in the so-called “Solvability Complexity Index Hierarchy” recently inroduced by
Hansen et al.
A note on the multiple fractional integrals defined on the product of
quasi-metric measure spaces
Tsira Tsanava, ts.tsanava@gtu.ge
Georgian Technical University, Georgia
Coauthor: Vakhtang Kokilashvili
A complete characterization of a vector-measure →−µ = (µ1, · · · , µn) governing the boundedness
of the multiple fractional integral operator
I−→γ f (x1, . . . , xn) = ··· f (y1, . . ., yn )dµ1(y1) · · · dµn(yn) , →−γ = (γ1, . . . , γn)
n (dj(xj, yj))1−γj
X1 Xn j=1
from one mixed norm Lebesgue space L−−→→µp to another one L−−→→µq is obtained, where (Xi, di, µi),
i = 1, . . . , n, are quasi-metric measure spaces (spaces of nonhomogeneous type).
Acknowledgement. The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
379
Computing Eigenvalues of the Laplacian on Rough Domains
Frank Rösler, roslerf@cardiff.ac.uk
Cardiff University, United Kingdom
Coauthor: Alexei Stepanenko
We discuss a recent work in which we prove a general Mosco convergence theorem for bounded
Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this
notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures
spectral convergence. A key element of the proof is the development of a novel, explicit
Poincaré-type inequality, which is of independent interest.
These results are applied construct a universal algorithm capable of computing the eigenval-
ues of the Dirichlet Laplacian on a wide class of rough domains. This immediately leads to new
classifications in the so-called “Solvability Complexity Index Hierarchy” recently inroduced by
Hansen et al.
A note on the multiple fractional integrals defined on the product of
quasi-metric measure spaces
Tsira Tsanava, ts.tsanava@gtu.ge
Georgian Technical University, Georgia
Coauthor: Vakhtang Kokilashvili
A complete characterization of a vector-measure →−µ = (µ1, · · · , µn) governing the boundedness
of the multiple fractional integral operator
I−→γ f (x1, . . . , xn) = ··· f (y1, . . ., yn )dµ1(y1) · · · dµn(yn) , →−γ = (γ1, . . . , γn)
n (dj(xj, yj))1−γj
X1 Xn j=1
from one mixed norm Lebesgue space L−−→→µp to another one L−−→→µq is obtained, where (Xi, di, µi),
i = 1, . . . , n, are quasi-metric measure spaces (spaces of nonhomogeneous type).
Acknowledgement. The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
379
