arXiv subjects

Marta Lewicka

Publications and source records attributed to Marta Lewicka.

At least 19 recordsLinked to original sources

arXiv2026

Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films

We prove that, for a given $C^{r,\beta}$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space $\mathbb{R}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,\alpha}$ for any $\alpha<\alpha_0=\min\{\frac{r+\beta}{2}, \frac{1}{1+d(d+1)/k}\}$. Our theorem recovers several previously known results as special cases. The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1, \frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available. Our threshold flexibility exponent $\alpha_0$ agrees with that previously obtained for the closely related Monge-Amp\`ere system. As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4\alpha_0}{\alpha_0+1}$.

math.AP↗

arXiv2026

Full flexibility of the Monge-Amp\`ere system in codimension $d_*-d+1$

We prove that $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Amp\`ere system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every H\"older exponent $\alpha<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [K\"allen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_* + 1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_*-d+1$.

math.AP↗

arXiv2025

Full flexibility of isometric immersions of metrics with low H\"older regularity in Poznyak theorem's dimension

A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $\omega\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in{C}^{r,\beta}$, an isometric immersion of regularity ${C}^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ for any $\alpha<\min\{\frac{r+\beta}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches ${C}^{1,1-}$ for $g\in{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity ${C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity ${C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity ${C}^{1,1-}$ obtained byt K\"allen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity ${C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Amp\`ere system.

math.DG↗

arXiv2025

The Monge-Amp\`ere system in dimension two is fully flexible in codimension two

We prove that every $\mathcal{C}^1(\bar\omega)$-regular subsolution of the Monge-Amp\`ere system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity $\mathcal{C}^{1,\alpha}(\bar\omega)$ for any $\alpha<\min\{1, \frac{s+\beta}{2}\}$, where $\mathcal{C}^{s,\beta}$ is the assumed regularity of the system's right hand side. This result suggests the full flexibility of Poznyak's theorem for isometric immersions of $2$d Riemannian manifolds into $\mathbb{R}^4$, and asserts it in the parallel setting of the Monge-Amp\`ere system.

math.AP↗

arXiv2025

The Monge-Ampere system in dimension two and codimension three

We revisit the convex integration constructions for the Monge-Amp\`ere system and prove its flexibility in dimension $d=2$ and codimension $k=3$, up to $\mathcal{C}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the obtained H\"older exponent $1-\frac{1}{\sqrt{5}}$ is larger than $1/2$ but it is not contained in the full flexibility up to $\mathcal{C}^{1,1}$ result. Previous various approaches, based on Kuiper's corrugations, always led to the H\"older regularity not exceeding $\mathcal{C}^{1,1/2}$, while constructions based on the Nash spirals (when applicable) led to the regularity $\mathcal{C}^{1,1}$. Combining the two approaches towards an interpolation between their corresponding exponent ranges has been so far an open problem.

math.AP↗

arXiv2024

The Monge-Ampere system in dimension two: a further regularity improvement

We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, whereas presently we achieve flexibility up to $\mathcal{C}^{1,1}$ when $k\geq 4$ and up to $\mathcal{C}^{1,\frac{2^k-1}{2^{k+1}-1}}$ for any $k$. This first result uses the approach of K\"allen, while the second result iterates on the approach of Cao-Hirsch-Inauen and agrees with it for $k=1$ at the H\"older regularity up to $\mathcal{C}^{1,1/3}$.

math.AP↗

arXiv2023

Isometric immersions and applications

We provide an introduction to the old-standing problem of isometric immersions. We combine a historical account of its multifaceted advances, which have fascinated geometers and analysts alike, with some of the applications in the mathematical physics and mathematical materials science, old and new.

math.DG↗

arXiv2023

The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension

We prove a convex integration result for the Monge-Ampere system in dimension $d=2$ and arbitrary codimension $k\geq 1$. We achieve flexibility up to the Holder regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, improving hence the previous $\mathcal{C}^{1,\frac{1}{1+ 6/k}}$ regularity that followed from flexibility up to $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}}$ in our previous work, valid for any $d,k\geq 1$. The present result agrees with flexibility up to $\mathcal{C}^{1,\frac{1}{5}}$ for $d=2, k=1$ obtained by Conti, Delellis, Szekelyhidi, as well as with the $\mathcal{C}^{1,\alpha}$ result where $\alpha\to 1$ as $k\to\infty$, due to Kallen.

math.AP↗

arXiv2022

The Monge-Ampere system: convex integration in arbitrary dimension and codimension

In this paper, we study flexibility of weak solutions to the Monge-Amp\`ere system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Amp\`ere equation in $d=2$ dimensions, naturally arising from the prescribed curvature problem and closely related to the classical problem of isometric immersions (II). Our main result achieves density in the set of subsolutions, of the H\"older $\mathcal{C}^{1,\alpha}$ solutions to the Von K\'arm\'an system (VK) which is the weak formulation of (MA). The regularity exponent $\alpha$ is any exponent satisfying $\alpha<\frac{1}{1+ d(d+1)/k}$ where $d$ is an arbitrary dimension and $k$ an arbitrary codimension of the problem. At $k=1$, this agrees with the regularity $\mathcal{C}^{1,\alpha}$ for (II) with any $\alpha <\frac{1}{1+d(d+1)}$, proved by Conti, Delellis and Szekelyhidi. At $d=2, k=1$, this extends the initial findings by the author and Pakzad for (MA). Our result seems to be optimal, from the technical viewpoint, for the corrugation-based convex integration scheme. In particular, it covers the codimension interval $k\in \big(1, d(d+1)\big)$ so far uncharted even for the system (II), since the regularity $\mathcal{C}^{1,\alpha}$ with any $\alpha <1$ achieved by K\"allen in \cite{Kallen}, strictly requires a large codimension. Our second main result reproduces K\"allen's result in the context of (MA), obtaining density in the set of subsolutions, of $\mathcal{C}^{1,\alpha}$ regular solutions for any $\alpha<1$ whenever $k\geq d(d+1)$. As an application of our results for (VK), we derive an energy scaling bound in the quantitative immersability of Riemannian metrics, for nonlinear energy functionals modelled on the energies of deformations of thin prestrained films in the nonlinear elasticity.

math.AP↗

arXiv2021

Geometric mechanics of random kirigami

The presence of cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields simple scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. By varying the shape and size of the geodesic in a kirigamized sheet, we show that we can control the deployment trajectory of the sheet, and thence its functional properties as a robotic gripper or a soft light window. Overall our study of random kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts.

cond-mat.soft↗

arXiv2021

Geometry, Analysis and Morphogenesis: Problems and Prospects

The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.

math.GM↗

arXiv2021

Geodesics and isometric immersions in kirigami

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.

math.DG↗

arXiv2020

Dimension reduction for thin films prestrained by shallow curvature

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the F\"oppl-von K\'arm\'an equations, we find the $\Gamma$-limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff-Love extensions of the highly perturbative, H\"older-regular solutions to the Monge-Ampere equation obtained by means of convex integration.

math.AP↗

arXiv2020

On asymptotic expansions for the fractional infinity Laplacian

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $\Delta_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65}, pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $\epsilon$, and allow for the identification of $\Delta_\infty^s\phi(x)$ as the $\epsilon^{2s}$-order coefficient of the deviation of the $\epsilon$-average from the value $\phi(x)$, in the limit $\epsilon\to 0+$. The averages are well posed for functions $\phi$ that are only Borel regular and bounded.

math.AP↗

arXiv2020

Non-local tug-of-war with noise for the geometric fractional $p$-Laplacian

This paper concerns the fractional $p$-Laplace operator $\Delta_p^s$ in non-divergence form, which has been introduced in [Bjorland, Caffarelli, Figalli (2012)]. For any $p\in [2,\infty)$ and $s\in (\frac{1}{2},1)$ we first define two families of non-local, non-linear averaging operators, parametrised by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb{R}^N\to \mathbb{R}$. We prove that $\Delta_p^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb{R}^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_p^s$, when posed in a domain $\mathcal{D}$ that satisfies the external cone condition and subject to bounded, uniformly continuous data on $\mathbb{R}^N\setminus \mathcal{D}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on $p$ and $N$, and whose $\epsilon$-tip has been removed.

math.AP↗

arXiv2020

A Course on Tug-of-War Games with Random Noise

This is a preprint of Chapter 2 in the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, reproduced with permission of Springer Nature Switzerland AG. We present the basic relation between the linear potential theory and random walks. This fundamental connection, developed by Ito, Doob, Levy and others, relies on the observation that harmonic functions and martingales share a common cancellation property, expressed via mean value properties. It turns out that, with appropriate modifications, a similar observation and approach can be applied also in the nonlinear case, which is of main interest in our Course Notes. Thus, the present Chapter serves as a stepping stone towards gaining familiarity with more complex nonlinear constructions. After recalling the equivalent defining properties of harmonic functions, we introduce the ball walk. This is an infinite horizon discrete process, in which at each step the particle, initially placed at some point $x_0$ in the open, bounded domain $\mathcal{D}\subset\mathbb{R}^N$, is randomly advanced to a new position, uniformly distributed within the following open ball: centered at the current placement, and with radius equal to the minimum of the parameter $\epsilon$ and the distance from the boundary $\partial\mathcal{D}$. With probability one, such process accumulates on $\partial\mathcal{D}$ and $u^\epsilon(x_0)$ is then defined as the expected value of the given boundary data $F$ at the process limiting position. Each function $u^\epsilon$ is harmonic, and if $\partial\mathcal{D}$ is regular, then each $u^\epsilon$ coincides with the unique harmonic extension of $F$ in $\mathcal{D}$. One sufficient condition for regularity is the exterior cone condition.

math.AP↗

arXiv2020

Lipschitz regularity of graph Laplacians on random data clouds

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the $L^2$-convergence rates established by two of the authors in previous work.

math.AP↗

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 $\Omega^h=\omega\times (-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $\Omega^1$, 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 $G_{\mid \Omega^h}$ on $\Omega^h,$ 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 $\Gamma$-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 $h\to 0$.

math.AP↗