arXiv subjects

Yury Grabovsky

Publications and source records attributed to Yury Grabovsky.

At least 19 recordsLinked to original sources

arXiv2026

Elastodynamics from a variational standpoint: integral equalities and inequalities

As it is well known, solutions of equations of nonlinear elastodynamics, representing extremals of the action functional, can form shocks. We adapt the classical approach of Emmy Noether to such singular extremals and derive the appropriately generalized integral relations within Calculus of Variations. We apply them to elastodynamical extremals with shocks, obtaining new integral relations involving kinetic and elastic energies. For the extremals representing thermodynamically admissible (entropy) solutions of the corresponding hyperbolic Euler-Lagrange equations, the classical equalities, characterizing the variational approach of Noether, expectedly transform into ineqalities. We show that, rather remarkably, despite the crucial role of material velocity in the fully inertial energy redistribution processes, the corresponding kinetic energy can be completely eliminated from the expression for the dynamically stored elastic energy, even in the presence of shocks.

math-ph↗

arXiv2026

Clapeyron-type theorems in nonlinear elasticity

Clapeyron's Theorem of classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. These relations reflect specific properties of Lagrangians, that we call ``partial variational symmetries'', since they are more general than classical variational symmetries. In the framework of nonlinear elasticity, partial variational symmetries, such as scale invariance, or scaling homogeneity, lead, via Noether's analysis to different nonlinear generalizations of Clapeyron's Theorem that combine naturally the work of physical and configurational forces. We present a series of illuminating examples showing the effectiveness of the obtained general results in different problems of nonlinear elasticity.

math-ph↗

arXiv2025

Generalized Clapeyron's theorem

Clapeyron's Theorem in classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. In the framework of nonlinear elasticity these relations reflect various partial symmetries of the material response, for instance, scale-invariance or scaling homogeneity. In particular, when the energy functional is scale-free, the obtained result can be interpreted as the Generalized Clapeyron's Theorem (GCT). Remarkably, it combines rather naturally the work of physical and configurational forces. We present a series of illuminating case studies showing the variety of applications of various obtained relations in different seemingly unrelated problems of mechanics.

math-ph↗

arXiv2024

Complete characterization of symmetric Kubo-Ando operator means satisfying Moln\'ar's weak associativity

We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in L. Moln\'ar, ``Characterizations of certain means of positive operators," Linear Algebra Appl. 567 (2019) 143-166, where he gives a characterization of this subclass (which we call the Moln\'ar class of means) in terms of the properties of their representing operator monotone functions. Moln\'ar's paper leaves open the problem of determining if the geometric mean is the only such mean in that subclass. Here we give a negative answer to this question by constructing an order-preserving bijection between this class and a class of real measurable odd periodic functions bounded in absolute value by $1/2$. Each member of the latter class defines a Molnar mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Moln\'ar class and construct several infinite families of explicit examples of Moln\'ar means that are not the geometric mean. Our analysis also shows how to modify Moln\'ar's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.

math.FA↗

arXiv2024

On feasibility of extrapolation of completely monotone functions

The feasibility of extrapolation of completely monotone functions can be quantified by examining the worst case scenario, whereby a pair of completely monotone functions agree on a given interval to a given relative precision, but differ as much as it is theoretically possible at a given point. We show that extrapolation is impossible to the left of the interval, while the maximal discrepancy to the right exhibits a power law typical for extrapolation of similar classes of complex analytic functions. The power law exponent is derived explicitly, and shows a precipitous drop immediately beyond the right end-point, with a subsequent decay to zero inversely proportional to the distance from the interval. The local extrapolation problem, where the worst discrepancy from a given completely monotone function is sought, is also analyzed. In this case explicit and easily verifiable optimality conditions are derived, enabling us to solve the problem exactly for a single decaying exponential. In the general case, our approach leads to a natural algorithm for computing solutions to the local extrapolation problem numerically. The methods developed in this paper can easily be adapted to other classes of analytic functions represented as integral transforms of positive measures with analytic kernels.

math.CV↗

arXiv2023

Rigidity-induced critical points

While classical theory of phase transitions deals with systems where shape variation is energetically neutral, the account of rigidity can lead to the emergence of new thermodynamic features. One of them is a special type of critical points that are characteristic of phase transitions specifically in solids. We develop a general theory of such rigidity-induced critical points and illustrate the results by analyzing in detail the case of an isotropic, geometrically nonlinear solid undergoing a volumetric phase transition at zero temperature.

cond-mat.mtrl-sci↗

arXiv2023

A class of nonlinear elasticity problems with no local but many global minimizers

We present a class of models of elastic phase transitions with incompatible energy wells in any space dimension, where an abundance of Lipschitz global minimizers in a hard device coexists with a complete lack of strong local minimizers. The analysis hinges on the proof that every strong local minimizer in a hard device is also a global minimizer which is applicable much beyond the chosen class of models. Along the way we show that a new proof of sufficiency for a subclass of affine boundary conditions can be built around a novel nonlinear generalization of the classical Clapeyron theorem, whose subtle relation to dynamics was studied extensively by R. Fosdick.

math.AP↗

arXiv2023

Solid phase transitions in the liquid limit

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a ``near-liquid'' limit. In the language of calculus of variations we probe limits of quasiconvexity in an ``almost liquid'' solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope.

math-ph↗

arXiv2023

A theory of inductive loops in electrochemical impedance spectroscopy

We demonstrate that failure of time-invariance assumption in the modeling of electrochemical systems by equivalent circuits can lead to the formation of low frequency "inductive loops" that manifest themselves as positive imaginary parts of the impedance function. Assuming that the properties of the equivalent circuits change slowly in time we perform an asymptotic analysis and obtain a new integral representation of the impedance function that reduces to the standard one at high frequencies, while exhibiting inductive loops at low frequencies.

physics.chem-ph↗

arXiv2021

On the commutation properties of finite convolution and differential operators I: commutation

Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to complex-valued and non self-adjoint operators. In addition, we introduce a new kind of commutation relation, which we call sesquicommutation, that also has implications for the spectral properties of the integral operator. In this case we also compute a complete list of sequicommuting pairs of integral and differential operators.

math.CA↗

arXiv2021

Reconstructing Stieltjes functions from their approximate values: a search for a needle in a haystack

Material response of real, passive, linear, time-invariant media to external influences is described by complex analytic functions of frequency that can always be written in terms of Stieltjes functions -- a special class of analytic functions mapping complex upper half-plane into itself. Reconstructing such functions from their experimentally measured values at specific frequencies is one of the central problems that we address in this paper. A definitive reconstruction algorithm that produces a certificate of optimality as well as a graphical representation of the uncertainty of reconstruction is proposed. Its effectiveness is demonstrated in the context of the electrochemical impedance spectroscopy.

math.OC↗

arXiv2020

Exact relations and links for two-dimensional thermoelectric composites

This is a report of the massive multi-year effort by the author and two graduate students Huilin Chen and Sarah Childs to compute all exact relations and links for two-dimensional thermoelectric composites. At the moment I have no time to prepare a proper "archival quality" manuscript with a good introduction and references. However, I believe that the results, concisely summarized in the last three sections of this report, should be made available to the research community even in this unfinished form.

math.AP↗

arXiv2020

On feasibility of extrapolation of the complex electromagnetic permittivity function using Kramer-Kronig relations

We study the degree of reliability of extrapolation of complex electromagnetic permittivity functions based on their analyticity properties. Given two analytic functions, representing extrapolants of the same experimental data, we examine how much they can differ at an extrapolation point outside of the experimentally accessible frequency band. We give a sharp upper bound on the worst case extrapolation error, in terms of a solution of an integral equation of Fredholm type. We conjecture and give numerical evidence that this bound exhibits a power law precision deterioration as one moves further away from the frequency band containing measurement data.

math-ph↗

arXiv2019

Analytic continuation in an annulus and in a Bernstein ellipse

Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In fact, well known theorems, guarantee that every continuous function can be uniformly approximated by analytic functions (polynomials or rational functions, for example). We consider several analytic continuation problems with typical global boundedness constraints. All such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the `worst case' function as a solution of a linear equation with a compact operator. In special geometries, such as the circular annulus this equation can be solved explicitly. The obtained solution is then used to determine the power law exponent for the analytic continuation from an interval between the foci of a Bernstein ellipse to the entire ellipse. In those cases where such exponents have been determined in prior work our results reproduce them faithfully.

math.AP↗

arXiv2019

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.

math.CV↗

arXiv2019

On the commutation properties of finite convolution and differential operators II: sesquicommutation

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization of commuting pairs $KL=LK$ and provides an exhaustive list of kernels admitting commuting or sesquicommuting differential operators.

math.AP↗

arXiv2018

Optimal error estimates for analytic continuation in the upper half-plane

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to provide a quantitative version of this statement. Given that $f$ from a unit ball in $H^2$ is small on $\Gamma$ (say, its $L^2$ norm is of order $\epsilon$), how does this affect the magnitude of $f$ at a point $z$ away from the curve? When $\Gamma \subset \partial \mathbb{H}_+$, we give a sharp upper bound on $|f(z)|$ of the form $\epsilon^\gamma$, with an explicit exponent $\gamma=\gamma(z) \in (0,1)$ and describe the maximizer function attaining the upper bound. When $\Gamma \subset \mathbb{H}_+$ we give an implicit sharp upper bound in terms of a solution of an integral equation on $\Gamma$. We conjecture and give evidence that this bound also behaves like $\epsilon^\gamma$ for some $\gamma=\gamma(z) \in (0,1)$. These results can also be transplanted to other domains conformally equivalent to the upper half-plane.

math.AP↗

arXiv2018

On the commutation properties of finite convolution and differential operators I: commutation

The commutation relation $KL = LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.

math.AP↗