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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.04235 |
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| _version_ | 1866929329617764352 |
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| author | Wang, Ling Zhou, Bin |
| author_facet | Wang, Ling Zhou, Bin |
| contents | In this paper, we establish the interior $C^{1,α}$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann (\textit{J. Econom. Theory} \textbf{146} (2011), no. 2, 454-478). The $C^{1,1}$ regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang(\textit{arXiv:2303.04937v3}). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions to the principal-agent problems with $q$-power cost are $C^{1,\frac{1}{q-1}}$ when $q > 2$ and $C^{1,1}$ when $1<q\leq 2$. Examples can show that this regularity is optimal when $q\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $C^{1,α}$ regularity of variational problems with a convexity constraint Wang, Ling Zhou, Bin Analysis of PDEs In this paper, we establish the interior $C^{1,α}$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann (\textit{J. Econom. Theory} \textbf{146} (2011), no. 2, 454-478). The $C^{1,1}$ regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang(\textit{arXiv:2303.04937v3}). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions to the principal-agent problems with $q$-power cost are $C^{1,\frac{1}{q-1}}$ when $q > 2$ and $C^{1,1}$ when $1<q\leq 2$. Examples can show that this regularity is optimal when $q\geq 2$. |
| title | $C^{1,α}$ regularity of variational problems with a convexity constraint |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.04235 |