Page 642 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 642
MATHEMATICAL PHYSICS
2 + nm ≤ q < 2, max qi < q 1 , 1 1n1
1+n 1+ =.
0≤i≤n
n q n i=1 qi
The main difficulty lies in the fact that part of mi < 1 (singular case), and another part of
mi > 1 (degenerate case). We found a universal approach to study the properties of solutions
of the anisotropic porous medium equation which not depends on the values of the anisotropic
exponents mi. We established the pointwise condition for removability of the singularity for so-
lutions of the equation (1) [1]. We also obtained the pointwise estimates of solutions, depending
on the relations between the exponents mi and qi (for the equation (3) [3]), mi and q ( for the
equation (2) in case f (u) = uq [2]) which guarantee that the point singularity is removable. The
proof of removability result is based on the new a priori estimates of "large" type solutions. In
particular, we obtain the Keller-Osserman type estimate of the solution to the problems (2), (4)
and (3), (4).
References
[1] M.A. Shan, Removable isolated singularities for solutions of anisotropic porous medium
equation. Annali di Matematica Pure ed Applicata 196(5), 1913–1926 (2017)
[2] M.A. Shan, Keller-Osserman a priori estimates and removability result for the anisotropic
porous medium equation with absorption term. Ukrainian Mathematical Bulletin, 15(1),
80 – 93 (2018)
[3] M.A. Shan, I.I. Skrypnik, Keller-Osserman estimates and removability result for the
anisotropic porous medium equation with gradient absorption term. Mathematische
Nachrichten, (in press)
Non-periodic ground states of one-dimensional, non-frustrated, two-body
interactions
Jacek Mie¸kisz, miekisz@mimuw.edu.pl
University of Warsaw, Poland
Since the discovery of quasicrystals, one of the problems in statistical mechanics is to construct
microscopic models of interacting atoms or molecules in which all ground-state configurations
minimizing energy are non-periodic.
We construct for the first time examples of one-dimensional classical lattice–gas models
with non-frustrated, two-body, infinite-range interactions and without periodic ground-state
configurations. Ground-state configurations of our models are Sturmian sequences defined by
irrational rotations on the circle. We present minimal sets of forbidden patterns which define
Sturmian sequences in a unique way. Our interactions assign positive energies to forbidden pat-
terns and are equal to zero otherwise. We illustrate our construction by the well-known example
of the Fibonacci sequences.
We will also discuss stability of one-dimensional non-periodic ground-state configurations
with respect to finite-range perturbations of interactions. We will show that they are not stable
for fast-decaying interactions.
References
[1] Aernout van Enter, Henna Koivusalo, and Jacek Mie¸kisz, Sturmian Ground States in Clas-
sical Lattice–Gas Models, Journal of Statistical Physics 178: 832–844 (2020).
640
2 + nm ≤ q < 2, max qi < q 1 , 1 1n1
1+n 1+ =.
0≤i≤n
n q n i=1 qi
The main difficulty lies in the fact that part of mi < 1 (singular case), and another part of
mi > 1 (degenerate case). We found a universal approach to study the properties of solutions
of the anisotropic porous medium equation which not depends on the values of the anisotropic
exponents mi. We established the pointwise condition for removability of the singularity for so-
lutions of the equation (1) [1]. We also obtained the pointwise estimates of solutions, depending
on the relations between the exponents mi and qi (for the equation (3) [3]), mi and q ( for the
equation (2) in case f (u) = uq [2]) which guarantee that the point singularity is removable. The
proof of removability result is based on the new a priori estimates of "large" type solutions. In
particular, we obtain the Keller-Osserman type estimate of the solution to the problems (2), (4)
and (3), (4).
References
[1] M.A. Shan, Removable isolated singularities for solutions of anisotropic porous medium
equation. Annali di Matematica Pure ed Applicata 196(5), 1913–1926 (2017)
[2] M.A. Shan, Keller-Osserman a priori estimates and removability result for the anisotropic
porous medium equation with absorption term. Ukrainian Mathematical Bulletin, 15(1),
80 – 93 (2018)
[3] M.A. Shan, I.I. Skrypnik, Keller-Osserman estimates and removability result for the
anisotropic porous medium equation with gradient absorption term. Mathematische
Nachrichten, (in press)
Non-periodic ground states of one-dimensional, non-frustrated, two-body
interactions
Jacek Mie¸kisz, miekisz@mimuw.edu.pl
University of Warsaw, Poland
Since the discovery of quasicrystals, one of the problems in statistical mechanics is to construct
microscopic models of interacting atoms or molecules in which all ground-state configurations
minimizing energy are non-periodic.
We construct for the first time examples of one-dimensional classical lattice–gas models
with non-frustrated, two-body, infinite-range interactions and without periodic ground-state
configurations. Ground-state configurations of our models are Sturmian sequences defined by
irrational rotations on the circle. We present minimal sets of forbidden patterns which define
Sturmian sequences in a unique way. Our interactions assign positive energies to forbidden pat-
terns and are equal to zero otherwise. We illustrate our construction by the well-known example
of the Fibonacci sequences.
We will also discuss stability of one-dimensional non-periodic ground-state configurations
with respect to finite-range perturbations of interactions. We will show that they are not stable
for fast-decaying interactions.
References
[1] Aernout van Enter, Henna Koivusalo, and Jacek Mie¸kisz, Sturmian Ground States in Clas-
sical Lattice–Gas Models, Journal of Statistical Physics 178: 832–844 (2020).
640
