论文标题
分裂型变异问题与线性生长条件
Splitting-type variational problems with linear growth conditions
论文作者
论文摘要
对于一类主要凸面拆分型能量密度$ f $ $ f $：$ \ mathbb {r}^2 \ to \ mathbb {r} $，\ [f（ξ_1，ξ_2）= f_1 = f_1 \ big big big（big big big big big big） \ big）\，\]，线性生长。作为主要结果，结果表明，无论相应的$ f_2 $如何，假设（$ t \ in \ mathbb {r} $）$ c_1（1+ | t |）^{ - μ_{1}} \ le f_1''（t）任何有限指数的$ \ partial_1 u $。我们还灌输了我们主要定理的一系列变体。我们最终注意到，在情况下，类似的结果$ f $：$ \ mathbb {r}^n \ to \ mathbb {r} $保持符号的明显更改。
Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: $\mathbb{R}^2 \to \mathbb{R}$, \[ f(ξ_1,ξ_2) = f_1\big( ξ_1 \big) + f_2\big( ξ_2 \big) \, , \] with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1 (1+|t|)^{-μ_{1}} \le f_1''(t) \le c_2\, ,\quad 1 < μ_1 < 2\, ,$ is sufficient to obtain higher integrability of $\partial_1 u$ for any finite exponent. We also inculde a series of variants of our main theorem. We finally note that similar results in the case $f$: $\mathbb{R}^n \to \mathbb{R}$ hold with the obvious changes in notation.
