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Davit Harutyunyan

Publications and source records attributed to Davit Harutyunyan.

At least 19 recordsLinked to original sources
arXiv2026

Korn and Poincaré-Korn inequalities for thin domains in $GSBD^p$

We prove Korn's first inequalities with and without boundary conditions in $GSBD^p$ ($1<p<\infty$) in $C^{1,1}$-regular thin domains with Lipschitz thickness boundaries with a constant $C h^{-p},$ that is optimal if the domain mid-surface contains a flat region, e.g., if the domain contains a piece of plate. For the one with boundary condition, we impose zero Dirichlet boundary conditions on the thin lateral faces of the shell, where we prove that the same estimate holds with no subtracted rigid motion like in the classical case for Sobolev vector fields.

math.AP↗
arXiv2026

On weighted rigidity estimates for bulk and thin domains

This work is concerned with weighted Geometric Rigidity Estimates and Korn's first inequalities in bulk and thin domains. We consider weights, that are a nonnegative power of the distance function to part of the boundary of the domain. For the case of thin domains, the distance is taken to be from the thin face of the domain boundary, and the constants depend on the domain thickness and when the mid-surface contains a flat region, the constants are proven to have optimal scaling as the thickness approaches zero. We employed some covering techniques utilized by Acosta, Cejas, and Duran in [\ref{bib:Aco.Cej.Dur.}] to prove a weighted Poincaré inequality, and later by Conti and Zwicknagl in [\ref{bib:Con.Zwi.}] to prove weighted Poincaré and classical Geometric Rigidity inequalities in Lipschitz domains. However, because the distance is taken to be only from part of the boundary, the covering parts become more delicate, especially for thin domains and the analysis becomes non-straightforward.

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arXiv2026

Rank-one convexity, polyconvexity, and extremality for three-dimensional elasticity tensors in various symmetry classes

It has been known that, for quadratic functions, quasiconvexity equals rank-one convexity but need not equal polyconvexity. While there is an explicit characterization of polyconvexity for quadratic energies, a similar characterization for quasiconvexity is not known in dimensions $n,N \geq 3.$ This paper is concerned with the search of extremal quasiconvex quadratic forms regarded as a linear elastic energy in dimensions $N=n=3$ in various symmetry classes of the elasticity tensor. In particular, we prove that for tensors with orthotropic symmetry, quasiconvexity implies polyconvexity, and thus the orthotropic class contains no nontrivial extremals. We show that this equivalence fails first at trigonal symmetry, where among some statemets, we provide a one-parameter family of non-polyconvex trigonal extreme rays. We also provide some general criteria for proving rank-one convexity, polyconvexity, and extremality, improving a result in [\ref{bib:Har.Hov.}].

math.AG↗
arXiv2022

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

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.

math.AP↗
arXiv2022

On the fractional Korn inequality in bounded domains: Counterexamples to the case $ps<1$

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 $ps>1$ for fractional $W^{s,p}_0(Ω)$ Sobolev fields in open and bounded $C^{1}$-regular domains $Ω\subset \mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain $Ω\subset \mathbb R^n$ 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 $ps>1$ 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 [\textit{Commun. Math. Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in the case $ps<1$ are interpolations of a constant affine rigid motion inside the domain away from the boundary, and of the zero field close to the boundary.

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arXiv2022

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

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 $h$ ($h>0$ 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 $h>0.$

math.AP↗
arXiv2021

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

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 $d>3,$ 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].

math.AG↗
arXiv2020

Rigidity of a thin domain depends on the curvature, width, and boundary conditions

This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A shallow thin domain is a thin domain that has in-plane dimensions of order $O(1)$ and $ε,$ where $ε\in (h,1)$ is a parameter (here $h$ is the thickness of the shell). The problem has been solved in [8,10] for the case $ε=1,$ with the outcome of the optimal constant $C\sim h^{-3/2},$ $C\sim h^{-4/3},$ and $C\sim h^{-1}$ for parabolic, hyperbolic and elliptic thin domains respectively. We prove in the present work that in fact there are two distinctive scaling regimes $ε\in (h,\sqrt h]$ and $ε\in (\sqrt h,1),$ such that in each of which the thin domain rigidity is given by a certain formula in $h$ and $ε.$ An interesting new phenomenon is that in the first (small parameter) regime $ε\in (h,\sqrt h]$, the rigidity does not depend on the curvature of the thin domain mid-surface.

math.AP↗
arXiv2020

The sharp $L^p$ Korn interpolation and second inequalities in thin domains

In the present paper we extend the $L^2$ Korn interpolation and second inequalities in thin domains, proven in [\ref{bib:Harutyunyan.4}], to the space $L^p$ for any $1<p<\infty.$ A thin domain in space is roughly speaking a shell with non-constant thickness around a smooth enough two dimensional surface. The inequality that we prove in $L^p$ holds for practically any thin domain $Ω\subset\mathbb R^3$ and any vector field $\Bu\in W^{1,p}(Ω).$ The constants in the estimate are asymptotically optimal in terms of the domain thickness $h.$ This in particular solves the problem of finding the asymptotics of the optimal constant in the classical Korn second inequality in $L^p$ for thin domains in terms of the domain thickness in almost full generality. The remarkable fact is that the interpolation inequality reduces the problem of estimating the gradient $\nabla\Bu$ in terms of the strain $e(\Bu)$ to the easier problem of estimating only the vector field $\Bu$, which is a Korn-Poincaré inequality.

math.AP↗
arXiv2019

The asymptotically sharp geometric rigidity interpolation estimate in thin bi-Lipschitz domains

This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke-James-Müller rigidity estimate [\textit{Comm. Pure Appl. Math.,} 55(11):1461-1506, 2002] for plates, our estimate holds for any proper rotations $\BR\in SO(3).$ Namely, the estimate bounds the $L^p$ distance of the gradient of any $\By\in W^{1,p}(Ω,\mathbb R^3)$ field from any constant proper rotation $\BR\in SO(3)$, in terms of the average $L^p$ distance (nonlinear strain) of the gradient $\nabla\By$ from the rotation group $SO(3)$, and the average $L^p$ distance of the field itself from the set of rigid motions corresponding to the rotation $\BR$. There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces the problem of estimating the gradient $\nabla\By$ in terms of the prototypical nonlinear strain $\int_Ω\mathrm{dist}^p(\nabla\By(x),SO(3))dx$ to the easier problem of estimating only the vector field $\By$ in terms of the nonlinear strain without any loss in the constant scalings as the Ansätze suggest. The later will be a geometric rigidity Korn-Poincaré type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free.

math.AP↗
arXiv2019

A note on the extreme points of the cone of quasiconvex quadratic forms with orthotropic symmetry

We study the extreme points of the cone of quasiconvex quadratic forms with linear elastic orthotropic symmetry. We prove that if the determinant of the acoustic matrix of the associated forth order tensor of the quadratic form is an extremal polynomial, then the quadratic form is an extreme point of the cone in the same symmetry class. The extremality of polynomials and quadratic forms here is understood in classical convex analysis sense.

math.AP↗
arXiv2018

On the Korn interpolation and second inequalities in thin domains

We consider shells of non-constant thickness in three dimensional Euclidean space around surfaces which have bounded principal curvatures. We derive Korn's interpolation (or the so called first and a half (The inequality first introduced in [Gra.Har.1])) and second inequalities on that kind of domains for $\Bu\in H^1$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are asymptotically optimal in terms of the domain thickness $h,$ with the leading order constant having the scaling $h$ as $h\to 0.$ This is the first work that determines the asymptotics of the optimal constant in the classical Korn second inequality for shells in terms of the domain thickness in almost full generality, the inequality being fulfilled for practically all thin domains $Ω\in\mathbb R^3$ and all vector fields $\Bu\in H^1(Ω).$ Moreover, the Korn interpolation inequality is stronger than Korn's second inequality, and it reduces the problem of estimating the gradient $\nabla\Bu$ in terms of the symmetrized gradient $e(\Bu)$, in particular any linear geometric rigidity estimates for thin domains, to the easier problem of proving the corresponding Poincaré-like estimates on the field $\Bu$ itself.

math.AP↗
arXiv2018

Quantitative anisotropic isoperimetric and Brunn-Minkowski inequalities for convex sets with improved defect estimates

In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic in $n.$ The tools are the Brenier's mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

math.DG↗
arXiv2018

The asymptotically sharp Korn interpolation and second inequalities for shells

We consider shells in three dimensional Euclidean space which have bounded principal curvatures. We prove Korn's interpolation (or the so called first and a half\footnote{The inequality first introduced in [6]}) and second inequalities on that kind of shells for $\Bu\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are optimal in terms of the asymptotics in the shell thickness $h,$ having the scalings $h$ or $O(1).$ The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré type estimate with the symmetrized gradient on the right hand side. In particular this applies to linear geometric rigidity estimates for shells, i.e., Korn's fist inequality without boundary conditions.

math.AP↗
arXiv2017

Weighted asymptotic Korn and interpolation Korn inequalities with singular weights

In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants $K$ (Korn's constant) in the inequalities depend on the domain thickness $h$ according to a power rule $K=Ch^α,$ where $C>0$ and $α\in R$ are constants independent of $h$ and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics $h^α$ is optimal as $h\to 0.$ The choice of the weights is motivated by several factors, in particular a spacial case occurs when making Cartesian to polar change of variables in two dimensions.

math.AP↗
arXiv2017

On the $L^\infty-$maximization of the solution of Poisson's equation: Brezis-Gallouet-Wainger type inequalities and applications

For the solution of the Poisson problem with an $L^\infty$ right hand side \begin{equation*} \begin{cases} -Δu(x) = f (x) & \mbox{in } D, u=0 & \mbox{on } \partial D, \end{cases} \end{equation*} we derive an optimal estimate of the form $$ \|u\|_\infty\leq \|f\|_\infty σ_D(\|f\|_1/\|f\|_\infty), $$ where $σ_D$ is a modulus of continuity defined in the interval $[0, |D|]$ and depends only on the domain $D$. In the case when $f\geq 0$ in $D$ the inequality is optimal for any domain and for any values of $\|f\|_1$ and $\|f\|_\infty.$ We also show that $$ σ_D(t)\leqσ_B(t),\text{ for }t\in[0,|D|], $$ where $B$ is a ball and $|B|=|D|$. Using this optimality property of $σ,$ we derive Brezis-Galloute-Wainger type inequalities on the $L^\infty$ norm of $u$ in terms of the $L^1$ and $L^\infty$ norms of $f.$ The estimates have explicit coefficients depending on the space dimension $n$ and turn to equality for a specific choice of $u$ when the domain $D$ is a ball. As an application we derive $L^\infty-L^1$ estimates on the $k-$th Laplace eigenfunction of the domain $D.$

math.AP↗
arXiv2017

Gaussian curvature as an identifier of shell rigidity

In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn's first (linear geometric rigidity estimate) and second inequalities on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like $h,$ and if the Gaussian curvature is negative, then the Korn constant scales like $h^{4/3},$ where $h$ is the thickness of the shell. These results have classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke, James and Müller for plates [14] (where they show that the Korn constant in the nonlinear Korn's first inequality scales like $h^2$), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincaré inequality proven for "boundary-less" shells by Lewicka and Müller in [37] in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents $1$ and $4/3$ in the present work appear for the first time in any sharp geometric rigidity estimate.

math.AP↗