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Mohammad Reza Pakzad

Publications and source records attributed to Mohammad Reza Pakzad.

17 recordsLinked to original sources
arXiv2024

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

We prove that the image of an isometric embedding into ${\mathbb R}^3$ of a two dimensionnal complete Riemannian manifold $(Σ, g)$ without boundary is a convex surface provided both the embedding and the metric $g$ enjoy a $C^{1,α}$ regularity for some $α>2/3$ and the distributional Gaussian curvature of $g$ is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.

math.DG↗
arXiv2024

The geometry of $C^{1,α}$ flat isometric immersions

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little Hölder regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,α}$ regularity with $α> 2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge-Ampère equation analyzed by Lewicka and the second author.

math.AP↗
arXiv2024

The biharmonic optimal support problem

We establish a $Γ$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set $K$. In the regime of very small vertical forces we identify the $Γ$-limit as being the bi-harmonic energy, with Dirichlet condition on the gluing region $K$, following a general strategy by Friesecke, James, and Müller that we have to adapt in presence of the glued region. Then we introduce a shape optimization problem that we call "optimal support problem" and which aims to find the best glued plate. In this problem the bi-harmonic energy is optimized among all possible glued regions $K$ that we assume to be connected and for which we penalize the length. By relating the dual problem with Griffith almost-minimizers, we are able to prove that any minimizer is $C^{1,α}$ regular outside a set of Hausdorff dimension strictly less then one.

math.AP↗
arXiv2021

Fractional Sobolev isometric immersions of planar domains

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and Hölder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,\frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.

math.AP↗
arXiv2017

Approximation by mappings with singular Hessian minors

Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(Ω)$ belonging to the little Hölder class $c^{1,α}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,α}$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.

math.AP↗
arXiv2017

Finite distortion Sobolev mappings between manifolds are continuous

We prove that if $M$ and $N$ are Riemannian, oriented $n$-dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings $W^{1,n}(Ω,\mathbb{R}^n)$, where $Ω\subset\mathbb{R}^n$ is an open set. The case of mappings between manifolds is much more difficult.

math.CA↗
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

math.AP↗
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.

math.AP↗
arXiv2014

Sobolev spaces of isometric immersions of arbitrary dimension and codimension

We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min\{2k, n\}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Ampère equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1)\times (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min\{2k, n\}$.

math.AP↗
arXiv2014

Variational models for prestrained plates with Monge-Ampère constraint

We derive a new model for pre-strained thin films, which consists of minimizing a biharmonic energy of deformations $v\in W^{2,2}$ satisfying the Monge-Ampère constraint $\det\nabla^2v = f$. We further discuss multiplicity properties of the minimizers of this model, in some special cases.

math.AP↗
arXiv2014

Models for elastic shells with incompatible strains

The three-dimensional shapes of thin lamina such as leaves, flowers, feathers, wings etc, are driven by the differential strain induced by the relative growth. The growth takes place through variations in the Riemannian metric, given on the thin sheet as a function of location in the central plane and also across its thickness. The shape is then a consequence of elastic energy minimization on the frustrated geometrical object. Here we provide a rigorous derivation of the asymptotic theories for shapes of residually strained thin lamina with nontrivial curvatures, i.e. growing elastic shells in both the weakly and strongly curved regimes, generalizing earlier results for the growth of nominally flat plates. The different theories are distinguished by the scaling of the mid-surface curvature relative to the inverse thickness and growth strain, and also allow us to generalize the classical Föppl-von Kármán energy to theories of prestrained shallow shells.

math.AP↗
arXiv2013

The Monge-Ampere constrained elastic theories of shallow shells

Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $Γ$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell scales like $h^α$ and the applied forces scale like $h^{α+2}$, in the limit when $h\to 0$. The main analytical ingredients are two independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries.

math.AP↗
arXiv2013

Rigidity and regularity of co-dimension one Sobolev isometric immersions

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to be true if the Sobolev regularity is weaker than $W^{2,2}$.

math.AP↗
arXiv2012

Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells

We perform a detailed analysis of first order Sobolev-regular infinitesimal isometries on developable surfaces without affine regions. We prove that given enough regularity of the surface, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. We discuss the implications of this result for the elasticity of thin developable shells.

math.AP↗
arXiv2008

The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells

Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^β$ with $2<β<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.

math.AP↗
arXiv2008

Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity

We discuss the limiting behavior (using the notion of Γ-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h^4 (where h is the thickness of a shell), we derive a limiting theory which is a generalization of the von Kármán theory for plates.

math.FA↗