Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers

TL;DR

This paper proves partial regularity results for relaxed minimizers of quasiconvex functionals with $(p,q)$-growth, extending existing exponent ranges and allowing signed integrands, thus broadening the understanding of regularity in calculus of variations.

Contribution

It establishes $ ext{C}^ ext{infinity}$-partial regularity for relaxed minimizers under broad $(p,q)$-growth conditions, including signed integrands and maximal exponent ranges, without measure representation assumptions.

Findings

Proves $ ext{C}^ ext{infinity}$-partial regularity for relaxed minimizers.

Extends exponent range for $(p,q)$-growth to maximal meaningful values.

Includes signed integrands in the analysis, broadening applicability.

Abstract

We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity condition on $F$, our results include pointwise $(p,q)$-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise $(p,q)$-growth conditions, our results extend the previously known exponent range from Schmidt's foundational work (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $p=1$. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representations \`{a} la Fonseca & Mal\'{y} (Ann. Inst. Henri Poincar\'{e}, Anal. Non Lin\'{e}aire 14, 309-338 (1997)).