My general research interests are in calculus of variations and
geometric measure theory (CVGMT) and spectral theory (ST). More precisely:
existence and regularity in CVGMT:
free boundary problems (with or without elastic energy)
including boundary discontinuities such as partitions, internal cracks,
wetting, edge delaminations, debonding, brittle fractures;
(anisotropic and crystalline) minimal surfaces;
capillarity problems; area functional;
higher order problems related to perimeter (such as Willmore problem);
evolution in CVGMT:
geometric evolution problems
such as (anisotropic and crystalline) mean curvature flow, mean
curvature flow of partitions; quasitatic evolution problems of
elastic materials; nonlocal evolution problems;
discrete Schrodinger operators in ST:
coupling constant threshold phenomenon;
multiparticle lattice systems, location of essential spectrum,
existence of discrete spectrum, Efimov effect; asymptotics of
eigenvalues.
Publication list
Preprints
Shokhrukh Kholmatov, Paolo Piovano : Existence of minimizers for the SDRI model.
Preprint (2020),
arXiv:2006.06096.
Shokhrukh Kholmatov, Ahmad Khalkhuzhaev, Mardon Pardabaev: Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian.
Preprint (2019),
arXiv:1910.01369.