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Quasiconvexity and self-improving size estimates

Bogdan Raiţă

Abstract

We show that Müller's $L\log L$ bound $$F(Du)\geq 0,\,Du\in L^p_{\mathrm{loc}}(\mathbb{R}^n)\implies F(Du)\in L\log L_{\mathrm{loc}}(\mathbb{R}^n)$$ for $F =\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$. This contrasts similar Hardy space bounds which hold only for null Lagrangians.

Quasiconvexity and self-improving size estimates

Abstract

We show that Müller's bound for and holds for quasiconcave which are homogeneous of degree . This contrasts similar Hardy space bounds which hold only for null Lagrangians.

Paper Structure

This paper contains 4 theorems, 24 equations.

Key Result

Theorem 1

Let $F\colon\mathbb{R}^{m\times n}\to\mathbb{R}$ be quasiconcave and $p$-homogeneous for $p>1$. Then with the estimate where $\omega\Subset\tilde{\omega}\Subset\Omega$ are open sets. Moreover, there exists $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$ with an analogous estimate.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm:main']}