On the structure of 𝓐-free measures and applications

G De Philippis, F Rindler - Annals of Mathematics, 2016‏ - JSTOR
We establish a general structure theorem for the singular part of 𝓐-free Radon measures,
where 𝓐 is a linear PDE operator. By applying the theorem to suitably chosen differential …

[کتاب][B] Calculus of variations

F Rindler - 2018‏ - Springer
The calculus of variations has its roots in the first problems of optimality studied in classical
antiquity by Archimedes (ca. 287–212 BC in Syracuse, Magna Graecia) and Zenodorus (ca …

Korn–Maxwell–Sobolev inequalities for general incompatibilities

F Gmeineder, P Lewintan, P Neff - Mathematical Models and …, 2024‏ - World Scientific
We establish a family of coercive Korn-type inequalities for generalized incompatible fields
in the superlinear growth regime under sharp criteria. This extends and unifies several …

Limiting Sobolev inequalities for vector fields and canceling linear differential operators.

J Van Schaftingen - Journal of the European Mathematical Society (EMS …, 2013‏ - ems.press
The estimate D k− 1 u Ln/(n− 1)≤ A (D) u L1 is shown to hold if and only if A (D) is elliptic
and canceling. Here A (D) is a homogeneous linear differential operator A (D) of order k on …

Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions

F Gmeineder, P Lewintan, P Neff - Calculus of Variations and Partial …, 2023‏ - Springer
We characterise all linear maps A: R n× n→ R n× n such that, for 1≤ p< n, PL p∗(R n)≤ c (A
[P] L p∗(R n)+ Curl PL p (R n)) holds for all compactly supported P∈ C c∞(R n; R n× n) …

Potentials for -quasiconvexity

B Raiță - Calculus of Variations and Partial Differential …, 2019‏ - Springer
We show that each constant rank operator AA admits an exact potential BB in frequency
space. We use this fact to show that the notion of A A-quasiconvexity can be tested against …

Partial regularity for BV minimizers

F Gmeineder, J Kristensen - Archive for Rational Mechanics and Analysis, 2019‏ - Springer
We establish an ε ε-regularity result for the derivative of a map of bounded variation that
minimizes a strongly quasiconvex variational integral of linear growth, and, as a …

On the trace operator for functions of bounded 𝔸-variation

D Breit, L Diening, F Gmeineder - Analysis & PDE, 2020‏ - msp.org
We consider the space BV 𝔸 (Ω) of functions of bounded 𝔸-variation. For a given first-order
linear homogeneous differential operator with constant coefficients 𝔸, this is the space of L 1 …

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

A Arroyo-Rabasa, G De Philippis… - Advances in calculus of …, 2020‏ - degruyter.com
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 …

[HTML][HTML] Embeddings for A-weakly differentiable functions on domains

F Gmeineder, B Raiţă - Journal of Functional Analysis, 2019‏ - Elsevier
We prove that the inhomogeneous estimate of vector fields on balls in R n (∫ B| D k− 1 u|
n/(n− 1) dx)(n− 1)/n⩽ c (∫ B| A u|+| u| dx) for all u∈ C∞(B¯, RN) holds if and only if the …