具有两个调制系数和对数型增长的向量值双相泛函的正则性
Regularity for vectorial double phase functionals with two modulating coefficients and logarithmic-type growth
- Kyungpook National University（庆北国立大学）
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AI总结：
本文研究具有两个调制系数和对数型增长的向量值双相泛函，证明在系数满足适当连续条件下极小元及其梯度的局部Hölder正则性，方法结合Gehring型高阶可积性与比较论证。
AI中文摘要：
我们研究具有两个调制系数和对数型增长的向量值双相泛函的局部极小元。能量密度为$a(x)|Du|^p+b(x)|Du|^p\log(e+|Du|)$，其中非负系数$a(\cdot)$和$b(\cdot)$可能消失，但它们的和有界远离零。假设$a(\cdot)$一致连续且$b(\cdot)$满足消失的对数-Hölder连续条件，我们证明极小元对$(0,1)$中的每个指数都是局部Hölder连续的。若两个系数都是Hölder连续的，则极小元的梯度是局部Hölder连续的。证明结合了Gehring型高阶可积性与比较论证。
英文摘要：
We study vector-valued local minimizers of double phase functionals with two modulating coefficients and logarithmic-type growth. The energy density is $a(x)|Du|^p+b(x)|Du|^p\log(e+|Du|)$, where the nonnegative coefficients $a(\cdot)$ and $b(\cdot)$ may vanish but their sum is bounded away from zero. Assuming that $a(\cdot)$ is uniformly continuous and that $b(\cdot)$ satisfies a vanishing log-Hölder continuity condition, we prove that minimizers are locally Hölder continuous for every exponent in $(0,1)$. If both coefficients are Hölder continuous, then the gradient of a minimizer is locally Hölder continuous. The proof combines Gehring-type higher integrability with a comparison argument.
