Related papers: Dimensional estimates and rectifiability for measu…
We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower…
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a…
The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute…
This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its…
We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this…
We define rectifiability in $\mathbb{R}^{n}\times\mathbb{R}$ with a parabolic metric in terms of $C^1$ graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent…
We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti…
The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space.…
The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic…
In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are $\mathrm{L}^1$-functions with the property that the symmetric…
We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$…
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable".…
We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} \mu = \sigma, \] where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$ are vector measures and the polar…
The aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint and to describe some applications.
We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is…
This paper computationally obtains optimal bounded-weight, binary, error-correcting codes for a variety of distance bounds and dimensions. We compare the sizes of our codes to the sizes of optimal constant-weight, binary, error-correcting…
PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…
A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations -- such as coefficient identification in partial differential…
We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a…
We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds…