Partial regularity and higher integrability for A-quasiconvex variational problems
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913014013231104 |
|---|---|
| author | Li, Zhuolin Raiţă, Bogdan |
| author_facet | Li, Zhuolin Raiţă, Bogdan |
| contents | We prove that minimizers of variational problems on open sets $Ω\subset \mathbb{R}^n$ $$ \mbox{minimize}\quad \mathcal E(v)=\int_Ωf(v(x))\mathrm{d} x\quad\text{for } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathscr{A}$-quasiconvex in a suitable sense. We consider $p$-growth problems with $1<p<\infty$, linear constant rank PDE operators $\mathscr{A}$ on $\mathbb{R}^n$ between vector spaces $V$ and $W$, and Dirichlet boundary conditions, in the sense that admissible fields are of the form $v=v_0+φ$, with $\mathscr{A}$-free $φ\in C_c^\infty(Ω,V)$. Our analysis also covers the ``potentials case'' $$ \mbox{minimize}\quad \mathcal F(u)=\int_Ωf(\mathscr{B} u(x))\mathrm{d} x\quad\text{for } u\in u_0+ C_c^\infty(Ω,U), $$ where $\mathscr{B}$ is another linear constant rank PDE operator on $\mathbb{R}^n$ between vector spaces $U,V$. We also prove appropriate higher integrability of minimizers for both types of problems. In addition, our approach covers non-autonomous integrands $f(x,v(x))$ or $f(x,\mathscr{B} u(x))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial regularity and higher integrability for A-quasiconvex variational problems Li, Zhuolin Raiţă, Bogdan Analysis of PDEs 49N60, 35B65, 49J45, 28B05, 35E20 We prove that minimizers of variational problems on open sets $Ω\subset \mathbb{R}^n$ $$ \mbox{minimize}\quad \mathcal E(v)=\int_Ωf(v(x))\mathrm{d} x\quad\text{for } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathscr{A}$-quasiconvex in a suitable sense. We consider $p$-growth problems with $1<p<\infty$, linear constant rank PDE operators $\mathscr{A}$ on $\mathbb{R}^n$ between vector spaces $V$ and $W$, and Dirichlet boundary conditions, in the sense that admissible fields are of the form $v=v_0+φ$, with $\mathscr{A}$-free $φ\in C_c^\infty(Ω,V)$. Our analysis also covers the ``potentials case'' $$ \mbox{minimize}\quad \mathcal F(u)=\int_Ωf(\mathscr{B} u(x))\mathrm{d} x\quad\text{for } u\in u_0+ C_c^\infty(Ω,U), $$ where $\mathscr{B}$ is another linear constant rank PDE operator on $\mathbb{R}^n$ between vector spaces $U,V$. We also prove appropriate higher integrability of minimizers for both types of problems. In addition, our approach covers non-autonomous integrands $f(x,v(x))$ or $f(x,\mathscr{B} u(x))$. |
| title | Partial regularity and higher integrability for A-quasiconvex variational problems |
| topic | Analysis of PDEs 49N60, 35B65, 49J45, 28B05, 35E20 |
| url | https://arxiv.org/abs/2412.10363 |