Bogdan Raiță

Scuola Normale Superiore di Pisa | Normale

We prove that the critical embedding $W^{\mathbb{A},1}(B)\hookrightarrow W^{k-1,\frac{n}{n-1}}$ holds if and only if the $k$-homogeneous, linear differential operator $\mathbb{A}$ on $\mathbb{R}^n$ from $\mathbb{R}^N$ to $\mathbb{R}^m$ has finite dimensional null-space. Here $B$ is a ball in $\mathbb{R}^n$ and $W^{\mathbb{A},1}(B)$ denotes the spac...

We give a generalization of Dorronsoro’s theorem on critical L p \mathrm {L}^p -Taylor expansions for B V k \mathrm {BV}^k -maps on R n \mathbb {R}^n ; i.e., we characterize homogeneous linear differential operators A \mathbb {A} of k k th order such that D k − j u D^{k-j}u has j j th order L n / ( n − j ) \mathrm {L}^{n/(n-j)} -Taylor expansion a....

We show that each constant rank operator $\mathcal{A}$ admits an exact potential $\mathbb{A}$ in frequency space. We use this fact to show that the notion of $\mathcal{A}$-quasiconvexity can be tested against compactly supported fields.

We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},d\mu)} \leq c \|\mu\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\mathbb{R}^{n},d\mathscr{L}^{n})}$$ hold for vector fields $u\in C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differenti...

We prove that for elliptic and canceling linear differential operators $\mathbb{B}$ of order $n$ on $\mathbb{R}^n$, continuity of a map $u$ can be inferred from the fact that $\mathbb{B} u$ is a measure. We also prove strict continuity of the embedding of the space $\mathrm{BV}^{\mathbb{B}}(\mathbb{R}^n)$ of functions of bounded $\mathbb{B}$-variat...

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations...

We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality $$ \|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R...

We show that for constant rank partial differential operators $$\mathscr {A}$$ A whose wave cones are spanning, generalized Young measures generated by bounded sequences of $$\mathscr {A}$$ A -free measures can be characterized by duality with $$\mathscr {A}$$ A -quasiconvex integrands of linear growth. This includes a characterization of the conce...

In the context of image processing, we study a class of integral regularizers defined in terms of spatially inhomogeneous integrands that depend on general linear differential operators. Particularly, the spatial dependence is assumed to be only measurable. The setting is made rigorous by means of the theory of Radon measures and of suitable functi...

We prove that for α ∈ (d − 1,d), one has the trace inequality ∫ℝd|IαF|dν≤C|F|(ℝd)∥ν∥Md−α(ℝd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\int}_{\mathbb{R}^{d}} |I_...

For l-homogeneous linear differential operators A of constant rank, we study the implicationvj⇀vinXAvj→AvinW−lY}⇒F(vj)⇝F(v)inZ, where F is an A-quasiaffine function and ⇝ denotes an appropriate type of weak convergence. Here Z is a local L1-type space, either the space M of measures, or L1, or the Hardy space H1; X,Y are Lp-type spaces, by which we...

We present a systematic treatment of the theory of Compensated Compactness under Murat’s constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed integrands, extending aspects of the results of Fonseca–Müller. The null Lagrangians are an important class of signed integrands, since they are the weakly co...

We study quasiconvex quadratic forms on $n \times m$ matrices which correspond to nonnegative biquadratic forms in $(n,m)$ variables. We disprove a conjecture stated by Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) as well as Harutyunyan--Hovsepyan (Arch. Ration. Mech. Anal. 244, 2022) that extremality in the cone of quasiconvex quadrat...

Let $$d\ge 2$$ d ≥ 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$\begin{aligned} \Vert I_\alpha F\Vert _{{\dot{B}}^{0,1}_{d/(d-\alpha ),1}(\mathbb {R}^d;\mathbb {R}^k)} \le C \Vert F \Vert _{L^1(\mathbb {R}^d;\mathbb {R}^k)} \end{aligned}$$ ‖ I α F ‖ B ˙ d / ( d - α ) , 1 0 , 1 ( R d ; R k ) ≤ C ‖ F ‖ L 1 ( R d ;...

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs...

We give a short proof of the fact that each homogeneous linear differential operator $A$ of constant rank admits a homogeneous potential operator $B$, meaning that $\ker A(x)=\mathrm{im\,}B(x)$ for $x\neq 0$. We make some refinements of the original result and some related remarks.

In the context of image processing, given a $k$-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the regularizers are integrals of spatially inhomogeneous integrands with convex dependence on the differential operat...

We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. The compensation effect takes the form of nonlinear elliptic estimates, where constraining the values of the field to lie in a cone compensates for the lack of ellipticity of the differential operator. We give a series of new examples of thi...

We prove that the concentration effects arising from weakly-* convergent sequences of gradients of maps of bounded variation have gradient structure. This is in stark contrast with the corresponding oscillation phenomena.

We prove that for $\alpha \in (d-1,d]$, one has the trace inequality \begin{align*} \int_{\mathbb{R}^d} |I_\alpha F| \;d\nu \leq C |F|(\mathbb{R}^d)\|\nu\|_{\mathcal{M}^{d-\alpha}(\mathbb{R}^d)} \end{align*} for all solenoidal vector measures $F$, i.e., $F\in M_b(\mathbb{R}^d,\mathbb{R}^d)$ and $\operatorname{div}F=0$. Here $I_\alpha$ denotes the R...

In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ \|I_\alpha F\|_{\dot{B}^{0,1}_{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^k)} \leq C \|F \|_{L^1(\mathbb{R}^d;\mathbb{R}^k)} $$ for all $F \in L^1(\mathbb{R}^d;\mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint...

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. 88 (1971), no. 1, p. 229-305) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estim...

For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ in $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function and $\rightsquigarrow$ denotes an appropriate type of we...

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.

We prove that for elliptic and canceling linear partial differential operators B of order n on Rn, continuity of a map u can be inferred from the fact that Bu is a measure. We also prove strict continuity of the embedding of the space BVB(Rn) into the space of continuous functions vanishing at infinity. Here, BVB(Ω) denotes the space of vector fiel...

We show that for constant rank partial differential operators $\mathscr{A}$, generalized Young measures generated by sequences of $\mathscr{A}$-free measures can be characterized by duality with $\mathscr{A}$-quasiconvex integrands of linear growth.

In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator $ \mathbb{A}$ and differential constraint $ \mathcal{C}$ for solutions to \[ \mathbb{A}u=f\phantom{\rule{1em}{0ex}}\text{subject to}\phantom{\rule{1em}{0ex}}\mathcal{C}f=0\phantom{\rule{1em}{0ex}}\text{ in...

We present a systematic treatment of the theory of Compensated Compactness under Murat's constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed integrands, extending the results of Fonseca--M\"uller. The null Lagrangians are an important class of signed integrands, since they are the weakly continuous...

In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*} \mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} to satisfy the est...

We show that the inequality $$ \|D^{k-1}(u-\pi u)\|_{\mathrm{L}^{n/(n-1)}(\mathbb{R}^n)}\leq c\|\mathbb{B}(D) u\|_{\mathrm{L}^1(\mathbb{R}^n)} $$ holds for vector fields $u\in\mathrm{C}^\infty_c$ if and only if $\mathbb{B}$ is canceling. Here $\pi$ denotes the $\mathrm{L}^2$-orthogonal projection onto the kernel of the $k$-homogeneous differential...

We prove that functions of locally bounded deformation on $\mathbb{R}^n$ are $\mathrm{L}^{n/(n-1)}$-differentiable almost everywhere. More generally, we show that this critical $\mathrm{L}^p$-differentiability result holds for functions of locally bounded $\mathbb{A}$-variation, provided that the first order, homogeneous, linear differential operat...