Regularity for minimizers with positive Jacobian

ABSTRACT

We deal with maps $u:\mathrm{\Omega }\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ minimizing variational integrals ${\int }_{\mathrm{\Omega }}\left[|Du\right(x){|}^p+h(detDu\left(x\right)\left)\right]\mathrm{d}x$, where $2\le p<n$, $h:(0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is convex and blows when $detDu\to 0^{+}$: $\underset{t\to 0^{+}}{lim}h\left(t\right)=+\mathrm{\infty }$. If such a blow up is a power of $|\ln \left(t\right)|$, then we derive regularity for the minimizer $u=(u^1,\dots ,u^n)$. We are able also to deal with integrals containing all the minors: ${\int }_{\mathrm{\Omega }}\left[|Du\right(x){|}^p+\sum _{s=2}^{n-1}|{\text{adj}}_s(Du\left(x\right)){|}^{q_s}+h(detDu\left(x\right)\left)\right]\mathrm{d}x$ with $q_s\ge 1$.

COMMUNICATED BY:

• Siegfried Carl

KEYWORDS:

• Minimizer
• regularity
• polyconvex
• nonlinear elasticity
• positive Jacobian

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49N60
• 35J60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

We acknowledge the support of MIUR, GNAMPA, INdAM, UNIVAQ and NSFC [10371050].