About

Davit Harutyunyan is a professor specializing in mathematical analysis with a focus on geometric rigidity estimates, Korn inequalities, and elasticity theory in thin domains and shells. His research extensively explores the mathematical foundations of elasticity, including the behavior of thin elastic structures such as cylindrical and elliptic shells under various conditions, with particular attention to buckling phenomena and the role of curvature. Harutyunyan has contributed to the development of sharp Korn interpolation and second inequalities, fractional Korn inequalities, and the characterization of extremal quasiconvex quadratic forms, advancing the understanding of rigidity and stability in elastic materials. His work also addresses the mathematical analysis of effective elasticity tensors in composite materials and the scaling laws relevant to magnetic thin films and ferromagnetic nanowires. Harutyunyan's research outputs include rigorous derivations of buckling load formulas, asymptotically sharp inequalities in thin domains, and the study of anisotropic isoperimetric inequalities, reflecting a deep engagement with both theoretical and applied aspects of elasticity and nonlinear analysis.

Research topics

• Combinatorics
• Physics
• Quantum mechanics
• Mathematical analysis
• Mathematics

Selected publications

• POSSIBLE APPLICATIONS OF ARTIFICIAL INTELLIGENCE IN THE SPHERE OF HEALTH INSURANCE

  SUSh Scientific Proceedings · 2024-12-30

  articleOpen accessSenior author

  Artificial intelligence (AI) and the rapid development of modern information technologies continue to impact all branches of the economy, including the insurance sector. Artificial intelligence in insurance is used for various purposes, from underwriting and pricing to claims processing, from fraud detection to customer service. This integration is evolving the industry, fundamentally changing how insurers assess risk, serve customers, and drive innovation in an industry historically characterized by risk aversion. Artificial intelligence is used in almost all insurance industries: health, auto, property, life, consumer, and more. The possibilities of using AI in health insurance are noteworthy: simplification of claims processing, fraud detection and prevention, personalized insurance premiums, and dynamic pricing. Within the framework of the article, the directions for using Artificial Intelligence in health insurance are discussed. As a result of studies, an attempt was made to automate the function of organizing customer service and health insurance tariff packages, for the implementation of which a Telegram bot was created. The bot works based on the Artificial Intelligence model, which was developed in the environment of the Python programming language. It enables the customer to receive a health insurance package with a personalized tariff plan after entering initial data online through a bot.

  PublisherDOI

• Fractional Korn’s inequalities without boundaryconditions

  Mathematics and Mechanics of Complex Systems · 2023-12-01 · 1 citations

  articleOpen access1st authorCorresponding

  Motivated by a linear nonlocal model of elasticity, this work establishes fractional analogues of Korn's first and second inequalities for vector fields in fractional Sobolev spaces defined over a bounded domain.The validity of the inequalities require no additional boundary condition, extending existing fractional Korn's inequalities that are only applicable for Sobolev vector fields satisfying zero Dirichlet boundary conditions.The domain of definition is required to have a C 1 -boundary or, more generally, a Lipschitz boundary with small Lipschitz constant.We conjecture that the inequalities remain valid for vector fields defined over any Lipschitz domain.We support this claim by presenting a proof of the inequalities for vector fields defined over planar convex domains.

  PublisherDOI

• On the fractional Korn inequality in bounded domains: Counterexamples to the case <i>ps</i>  &lt; 1

  Advances in Nonlinear Analysis · 2023 · 2 citations

  1st authorCorresponding
  • Combinatorics
  • Physics
  • Mathematics

  Abstract The validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mi>s</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> ps\gt 1 for fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {W}_{0}^{s,p}\left(\Omega ) Sobolev fields in open and bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {C}^{1} -regular domains <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> \Omega \subset {{\mathbb{R}}}^{n} . Also, in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mi>s</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:math> ps\lt 1 , for any open bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {C}^{1} domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> \Omega \subset {{\mathbb{R}}}^{n} , we construct counterexamples to the inequality, i.e., Korn’s first inequality fails to hold in bounded domains. The proof of the inequality in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mi>s</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> ps\gt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [ A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations , Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mi>s</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:math> ps\lt 1 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.

  PublisherOA PDFDOI

• The Buckling Load of Cylindrical Shells Under Axial Compression Depends on the Cross-Sectional Curvature

  Journal of Nonlinear Science · 2023-01-10 · 3 citations

  article1st authorCorresponding
  PublisherDOI

• The buckling load of cylindrical shells under axial compression depends on the cross-section curvature

  arXiv (Cornell University) · 2022-02-27

  preprintOpen access1st authorCorresponding

  It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter's formula predicts linear dependence of the buckling load $λ(h)$ of the shell thickness $h$ ($h&gt;0$ is a small parameter), one observes the dependence $λ(h)\sim h^{3/2}$ in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in [\textit{J. Nonl. Sci.,} Vol. 26, Iss. 1, pp. 83--119, Feb. 2016], that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling $λ(h)\sim h^{5/4},$ while shape imperfections are likely to result in the scaling $λ(h)\sim h^{3/2}.$ In this work we prove, that in fact the buckling load $λ(h)$ of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then $λ(h)\sim h,$ and when the the cross section curve has positive curvature except at finitely many points, then $C_1h^{8/5}\leq λ(h)\leq C_2h^{3/2}$ for $h$ small thickness $h&gt;0.$

  PublisherOA PDFDOI

• A Hint on the Localization of the Buckling Deformation at Vanishing Curvature Points on Thin Elliptic Shells

  Journal of Elasticity · 2022-11-29 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• On the Extreme Rays of the Cone of $$3\times 3$$ Quasiconvex Quadratic Forms: Extremal Determinants Versus Extremal and Polyconvex Forms

  Archive for Rational Mechanics and Analysis · 2022-02-07 · 2 citations

  article1st authorCorresponding
  PublisherDOI

• On the extreme rays of the cone of $3\times 3$ quasiconvex quadratic forms: Extremal determinats vs extremal and polyconvex forms

  arXiv (Cornell University) · 2021-02-15

  preprintOpen access1st authorCorresponding

  This work is concerned with the study of the extreme rays of the convex cone of $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). We characterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\cal C}_3;$ when the determinant is a perfect square, then the form is either an extreme ray of ${\cal C}_3$ or polyconvex; and finally, when the determinant is identically zero, then the form $f$ must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of ${\cal C}_3,$ and discuss about weak and strong etremals of ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of ${\cal C}_3$ do not hold for ${\cal C}_d$ for $d&gt;3,$ and thus case $d=3$ is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of ${\cal C}_3$ that were proved to be such. Our results also improve the ones proven by the first author and Milton [20].

  PublisherOA PDFDOI

• Rigidity of a Thin Domain Depends on the Curvature, Width, and Boundary Conditions

  Applied Mathematics & Optimization · 2021-02-16

  preprintOpen access
  PublisherOA PDFDOI

• A hint on the localization of the buckling deformation at vanishing curvature points on thin elliptic shells

  arXiv (Cornell University) · 2021-04-24

  preprintOpen access1st authorCorresponding

  The general theory of slender structure buckling by Grabovsky and Truskinovsky [\textit{Cont. Mech. Thermodyn.,} 19(3-4):211-243, 2007], (later extended in [\textit{Journal of Nonlinear Science.,} Vol. 26, Iss. 1, pp. 83--119, 2016] by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn's constant (in Korn's first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positive, negative, and zero (one principal curvature vanishing, and one apart from zero) Gaussian curvature shells, the optimal Korn constant in Korn's first inequality scales like the thickness to the power of $-1, -4/3,$ and $-3/2$ respectively. In this work we analyse the scaling of the optimal constant in Korn's first inequality for elliptic shells that contain a finite number of points where both principal curvatures vanish. We prove that the presence of at least one such point on the shell leads to the scaling drop from the thickness to the power of $-1$ to the thickness to the power of $-3/2.$ To our best knowledge, this is the first result in the direction for constant-sign curvature shells, that do not contain a developable region. In addition, under the assumption that a suitable trivial branch exists, we prove that in fact the buckling deformation of such shells under dead loading, should be localized at the vanishing curvature points, as the shell thickness h goes to zero.

  PublisherOA PDFDOI

Recent grants

• Rigidity and Buckling of Shells: Toward New Nonlinear Shell Theories

  NSF · $154k · 2018–2022

Frequent coauthors

• Graeme W. Milton

  15 shared

• Yury Grabovsky

  Temple University

  15 shared

• Hayk Mikayelyan

  University of Nottingham Ningbo China

  6 shared

• Narek Hovsepyan

  6 shared

• Trevor J. Dick

  University of Utah

  3 shared

• Justin Boyer

  University of Utah

  2 shared

• Andre Martins Rodrigues

  University of California, Santa Barbara

  2 shared

• Marc Briane

  Institut de recherche mathématique de Rennes

  2 shared