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The local Burkholder functional, quasiconvexity and Geometric Function Theory

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arxiv 2309.03495 v2 pith:3R2BPANF submitted 2023-09-07 math.AP math.CV

Kari Astala , Daniel Faraco , André Guerra , Aleksis Koski , Jan Kristensen This is my paper · ORCID
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keywords functionalsburkholderfunctiondeterminantfunctionalgeometriclocalnon-polyconvex
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abstract

We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasiconvexity for the Dacorogna--Marcellini Energy

    math.AP 2026-08 conditional novelty 8.0 of 10

    For the planar Dacorogna-Marcellini energy, quasiconvexity and rank-one convexity coincide with threshold |γ| ≤ 2/√3, and the same equivalence holds for all SO(2)×SO(2)-invariant quartic polynomials.

  2. A solution to Morrey's problem in $\mathbb{R}^{2\times m}$

    math.AP 2026-08 conditional novelty 8.0 of 10

    For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.

  3. Korn's inequality from the viewpoint of calculus of variations

    math.AP 2026-03 conditional novelty 8.0 of 10

    Korn's constant satisfies p^*-1 ≤ C(p,d) ≤ √3(p^*-1) in all dimensions, with exact p^*-1 for radial fields in dimension 2.

  4. Martingales, laminates and minimal Korn inequalities

    math.AP 2025-12 conditional novelty 8.0 of 10

    The minimal number of linear measurements controlling ∇u in Korn's second inequality is exactly 2d−1, and in the first inequality it grows like 2d asymptotically.

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