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Hayk Mikayelyan

Publications and source records attributed to Hayk Mikayelyan.

18 recordsLinked to original sources
arXiv2022

Stabilization technique applied to curve shortening flow in $\mathbb{R}^3$

We apply the stabilization technique, developed by T. Zelenyak in 1960s for parabolic equations, on curve shortening flow in $\R^3$, and derive several new monotonicity formulas. All of them share one main feature: the dependence of the "energy" term on the angle between the position vector and the plane orthogonal to the tangent vector. The first formula deals with the projection of the curve on the unit sphere, and computes the derivative of its length. The second formula is the generalization of the classical formula of G. Huisken, while the third one is the generalization of the monotonicity formula with logarithmic terms previously derived by the author for plane curves.

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.

math.AP↗
arXiv2021

Constrained optimal rearrangement problem leading to a new type obstacle problem

We consider a new type of obstacle problem in the cylindrical domain $Ω=D\times (0,1)$ arising from minimization of the functional $$ \int_Ω\frac{1}{2}|\nabla u|^2+χ_{\{v>0\}}udx, $$ where $v(x')=\int_0^1 u(x', t) dt $. We prove several existence and regularity results and show that the comparison principle does not hold for minimizers. This problem is derived from a classical optimal rearrangement problem in a cylindrical domain, under the constraint that the force function does not depend on the $x_n$ variable of the cylindrical axis. A mistake in the Theorem 4.2 of the previous version has been found. The statement remains an open problem.

math.AP↗
arXiv2020

The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford-Shah functional in the plain

We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, Γ)=\int_{B_1\backslashΓ} |\nabla u|^2 dx + λ^2 \fracπ{2} \mathcal{H}^1(Γ). $$ It is known that if the minimizer has a crack-tip in the ball $B_1$ (assume at the origin), then $u\approx λ\Im \sqrt{z}$ at this point. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation.

math.AP↗
arXiv2019

Regularity up to the Crack-Tip for the Mumford-Shah problem

We prove that if $(u,Γ)$ is a minimizer of the functional $$ J(u,Γ)=\int_{B_1(0)\setminus Γ}|\nabla u|^2dx +\H^1(Γ) $$ and $Γ$ connects $\partial B_1(0)$ to a point in the interior, then $Γ$ satisfies a point-wise $C^{2,α}$-estimate at the crack-tip. This means that the Mumford-Shah functional satisfies an additional, and previously unknown, Euler-Lagrange condition. ******* The previous version of the paper contained some mistakes, which has been fixed. More explanations/details has been added in Section 6.

math.AP↗
arXiv2017

Stationarity of the crack-front for the Mumford-Shah problem in 3D

In this paper we exhibit a family of stationary solutions of the Mumford-Shah functional in $\mathbb{R}^3$, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet \cite{b}. We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the crack-tip function for the Mumford-Shah problem in 2D is presented.

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↗
arXiv2015

Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.

math.NA↗
arXiv2014

Hopf's lemma for a class of singular/degenerate PDE-s

This paper concerns Hopf's boundary point lemma, in certain $C^{1,Dini}$-type domains, for a class of singular/degenerate PDE-s, including $p$-Laplacian. Using geometric properties of levels sets for harmonic functions in convex rings, we construct sub-solutions to our equations that play the role of a barrier from below. By comparison principle we then conclude Hopf's lemma.

math.AP↗
arXiv2010

Convexity of the free boundary for an exterior free boundary problem involving the perimeter

We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

math.AP↗