Marta Lewicka
YOU?
Author Swipe
View article: The Monge-Ampère system in dimension two is fully flexible in codimension two
The Monge-Ampère system in dimension two is fully flexible in codimension two Open
We prove that every $\mathcal{C}^1(\barω)$-regular subsolution of the Monge-Ampère system posed on a $2$-dimensional domain $ω$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity $\mathcal…
View article: The Monge-Ampere system in dimension two and codimension three
The Monge-Ampere system in dimension two and codimension three Open
We revisit the convex integration constructions for the Monge-Ampère 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 obt…
View article: The Monge-Ampere system in dimension two: a further regularity improvement
The Monge-Ampere system in dimension two: a further regularity improvement Open
We prove a convex integration result for the Monge-Ampère system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the Hölder regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, whereas p…
View article: The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension
The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension Open
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 $\…
View article: The Monge-Ampere system: convex integration in arbitrary dimension and codimension
The Monge-Ampere system: convex integration in arbitrary dimension and codimension Open
In this paper, we study flexibility of weak solutions to the Monge-Ampère system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Ampère equation in $d=2$ dimensions, naturally arising from the prescribed c…
View article: The mathematics of thin structures
The mathematics of thin structures Open
This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that…
View article: Geometry, analysis, and morphogenesis: Problems and prospects
Geometry, analysis, and morphogenesis: Problems and prospects Open
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, a…
View article: Geometric mechanics of random kirigami
Geometric mechanics of random kirigami Open
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 …
View article: Geometry, Analysis and Morphogenesis: Problems and Prospects
Geometry, Analysis and Morphogenesis: Problems and Prospects Open
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 an…
View article: Geodesics and isometric immersions in kirigami
Geodesics and isometric immersions in kirigami Open
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 enable…
View article: On asymptotic expansions for the fractional infinity Laplacian
On asymptotic expansions for the fractional infinity Laplacian Open
We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian [Formula: see text] for [Formula: see text]. This operator has been introduced and first studied in ( Comm. Pure …
View article: Dimension reduction for thin films prestrained by shallow curvature
Dimension reduction for thin films prestrained by shallow curvature Open
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 Föppl–von Kármán equations…
View article: Which domains have two-sided supporting unit spheres at every boundary point?
Which domains have two-sided supporting unit spheres at every boundary point? Open
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 contin…
View article: Visualization of the convex integration solutions to the Monge-Ampère equation
Visualization of the convex integration solutions to the Monge-Ampère equation Open
In this article, we implement the algorithm based on the convex integration result proved in [Lewicka-Pakzad Analysis and PDE (2017)] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approx- imating the anomalou…
View article: Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case
Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case Open
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…
View article: Thin Structures With Imposed Metric
Thin Structures With Imposed Metric Open
We consider thin structures with a non necessarily realizable imposed metric, that only depends on the surface variable. We give a unified presentation of the three main limit models. We establish the generalized membrane model and we show…
View article: The metric-restricted inverse design problem
The metric-restricted inverse design problem Open
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…