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 …

On rank one convex functions that are homogeneous of degree one

B Kirchheim, J Kristensen - Archive for rational mechanics and analysis, 2016‏ - Springer
We show that positively 1-homogeneous rank one convex functions are convex at 0 and at
matrices of rank one. The result is a special case of an abstract convexity result that we …

Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV

J Kristensen, F Rindler - Archive for rational mechanics and analysis, 2010‏ - Springer
Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys
108: 667–689, 1987) provide a quantitative tool for studying the one-point statistics of …

Characterization of Sobolev and BV spaces

G Leoni, D Spector - Journal of Functional Analysis, 2011‏ - Elsevier
Characterization of Sobolev and BV spaces Page 1 Journal of Functional Analysis 261 (2011)
2926–2958 www.elsevier.com/locate/jfa Characterization of Sobolev and BV spaces Giovanni …

Localization of nonlocal gradients in various topologies

T Mengesha, D Spector - Calculus of Variations and Partial Differential …, 2015‏ - Springer
In this paper, we study nonlocal gradients and their relationship to classical gradients. As the
nonlocality vanishes we demonstrate the convergence of nonlocal gradients to their local …

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 …

Quasiconvex Functionals of (pq)-Growth and the Partial Regularity of Relaxed Minimizers

F Gmeineder, J Kristensen - Archive for Rational Mechanics and Analysis, 2024‏ - Springer
We establish C∞-partial regularity results for relaxed minimizers of strongly quasiconvex
functionals F [u; Ω]:=∫ Ω F (∇ u) dx, u: Ω→ RN, subject to aq-growth condition| F (z)|≦ c …