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Marta Lewicka

Publications and source records attributed to Marta Lewicka.

At least 37 records · Page 2Linked to original sources
arXiv2019

Random walks and random tug of war in the Heisenberg group

We study the mean value properties of $\mathbf{p}$-harmonic functions on the first Heisenberg group $\mathbb{H}$, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres-Scheffield to provide the game-theoretical interpretation of the sub-elliptic $\mathbf{p}$-Laplacian; and of Manfredi-Parviainen-Rossi to characterize its viscosity solutions via the asymptotic mean value expansions.

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arXiv2019

Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case

We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits ($Γ$-limits) as $h\to 0$ in the film's thickness $h$. Firstly, we extend the prior results [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti] to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the "non-oscillatory" case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the "oscillatory" case), extending the results of [Agostiniani-Lucic-Lucantonio, Schmidt] to arbitrary metrics and higher order scalings. We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of $h$ and discuss the scaling regimes $h^2$ (Kirchhoff), $h^4$ (von Kármán) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [Lewicka]. Thirdly, we prove the coercivity inequalities for the singular limits at $h^2$- and $h^4$- scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling $h^4$.

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arXiv2018

Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

This paper concerns the variational description of prestrained materials, in the context of dimension reduction for thin films $Ω^h=ω\times (-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $Ω^1$, we study the question of what is the infimum of the averaged pointwise deficit of a given immersion from being an orientation-preserving isometric immersion of $G_{\mid Ω^h}$ on $Ω^h,$ over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $\mathcal{E}^h$, which can be seen as modifications of the classical nonlinear three-dimensional elasticity. Building on our previous results, we complete the scaling analysis of $\mathcal{E}^h$ and the derivation of $Γ$-limits of the scaled energies $h^{-2n}\mathcal{E}^h$, for all $n\geq 1$. We show the energy quantisation in the sense that the even powers $2n$ of $h$ are indeed the only possible ones (all of them are also attained). For each $n$, we identify the equivalent conditions for the validity of the corresponding scaling, in terms of the vanishing of appropriate Riemann curvatures of $G$ to certain orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $h\to 0$.

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arXiv2018

Which domains have two-sided supporting unit spheres at every boundary point?

We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain $Ω\subset\mathbb{R}^N$. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: "Those domains, whose unit normal is well defined and has Lipschitz constant one." We also offer an extension to infinitely dimensional spaces $L^p$, $p\in (1,\infty)$.

math.CA↗
arXiv2017

A Model of Controlled Growth

We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with $C^{2,α}$ boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion.

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arXiv2016

Convex integration for the Monge-Ampère equation in two dimensions

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration à la Nash & Kuiper and establish the related h-principle for the Monge-Ampère equation in two dimensions

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arXiv2016

The metric-restricted inverse design problem

We study a class of design problems in solid mechanics, leading to a variation on the classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new context, we derive a necessary and sufficient existence condition, given through a system of total differential equations, and discuss its integrability. In the classical context, the same approach yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor. In the present situation, the equations do not close in a straightforward manner, and successive differentiation of the compatibility conditions leads to a new algebraic description of integrability. We also recast the problem in a variational setting and analyze the infimum of the appropriate incompatibility energy, resembling the "non-Euclidean elasticity." We then derive a $Γ$-convergence result for the dimension reduction from $3$d to $2$d in the Kirchhoff energy scaling regime.

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arXiv2015

The obstacle problem for the $p$-laplacian via optimal stopping of Tug-of-War games

We present a probabilistic approach to the obstacle problem for for the $p$-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the step size go to zero, not unlike the case when Brownian motion is approximated by random walks. Rather than stopping the process when the boundary is reached, the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain.

math.AP↗
arXiv2015

A rigorous justification of the Euler and Navier-Stokes equations with geometric effects

We derive the 1D isentropic Euler and Navier-Stokes equations describing the motion of a gas through a nozzle of variable cross section as the asymptotic limit of the 3D isentropic Navier-Stokes system in a cylinder, the diameter of which tends to zero. Our method is based on the relative energy inequality satisfied by any weak solution of the 3D Navier-Stokes system and a variant of Korn-Poincare's inequality on thin channels that may be of independent interest.

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arXiv2015

Discrete approximations to the double-obstacle prtoblem, and optimal stopping of tug-of-war games

We study the double-obstacle problem for the p-Laplace operator, p 2 [2;1). We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-ofwar games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token's position, converges to 0. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.

math.AP↗
arXiv2015

Plates with incompatible prestrain of higher order

We study the effective elastic behaviour of the incompatibly prestrained thin plates, characterized by a Riemann metric $G$ on the reference configuration. We assume that the prestrain is "weak", i.e. it induces scaling of the incompatible elastic energy $E^h$ of order less than $h^2$ in terms of the plate's thickness $h$. We essentially prove two results. First, we establish the $Γ$-limit of the scaled energies $h^{-4}E^h$ and show that it consists of a von Kármán-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing $G_{2\times 2}$ (i.e. the prestrain metric on the midplate) in $\mathbb{R}^3$. Second, we prove that in the scaling regime $E^h\sim h^β$ with $β>2$, there is no other limiting theory: if $\inf h^{-2} E^h \to 0$ then $\inf E^h\leq Ch^4$, and if $\inf h^{-4}E^h\to 0$ then $G$ is realizable and hence $\min E^h = 0$ for every $h$.

math.AP↗
arXiv2015

On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions

We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets $Ω\subset \mathbb{R}^n$, and in the geometric rigidity estimate on the whole $\mathbb{R}^2$. We prove that the latter constant equals $\sqrt{2}$, and we discuss the relation of the former constants with the optimal Korn's constants under Dirichlet boundary conditions, and in the whole $\mathbb{R}^n$, which are well known to equal $\sqrt{2}$. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.

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arXiv2014

Plates with incompatible prestrain

We study the effective elastic behavior of incompatibly prestrained plates, where the prestrain is independent of thickness as well as uniform through the thickness. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric $G$ with the above properties, and seek the limiting behavior as the thickness goes to zero. Our results extand the prior analysis in M. Lewicka, M. R. Pakzad ESAIM Control Optim. Calc. Var. 17 (2011), no. 4. We first establish that the $Γ$-limit is a Kirchhoff type bending. Further, we show that the minimum energy configuration contains non-trivial Kirchhoff type bending -- i.e., the scaling of the three-dimensional energy is of the order of the cube of the plate thickness -- if and only if the Riemann curvatures $R^3_{112}, R^3_{221}$ and $ R_{1212}$ of $G$ do not identically vanish. We demonstrate through examples, the existence of a new regime where the three above curvatures of $G$ vanish (while the mid-plane of the plate may or may not be flat), but the limiting configuration still has energy that is of the order of Föppl - von Kármán plates. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for $G=\mbox{Id}_3 +γ\vec n\otimes \vec n$ given in terms of the inhomogeneous unit director field distribution $\vec n\in\mathbb{R}^3$.

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arXiv2014

Game Theoretical Methods in Nonlinear PDEs

Nonlinear PDEs, mean value properties, and stochastic differential games are intrinsically connected. In this short expository note, we will describe how the solutions to certain PDEs (of $p$-Laplacian type) can be interpreted as limits of values of a specific Tug-of-War game, when the step-size $ε$ determining the allowed length of move of a token, decreases to $0$.

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arXiv2014

On the variational limits of lattice energies on prestrained elastic bodies

We study the asymptotic behaviour of the discrete elastic energies in presence of the prestrain metric $G$, assigned on the continuum reference configuration $Ω$. When the mesh size of the discrete lattice in $Ω$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $Γ$-limit) energy. In case of the nearest-neighbour and next-to-nearest-neibghour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to $G$.

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