Abstract We prove sharp regularity results for minimizers of the functional P ( w , Ω ) : = ∫ Ω b ( x , w ) [ | D w | p + a ( x ) | D w | p log ⁡ ( e + | D w | ) ] d x , with w ∈ W 1 , 1 ( Ω ) , p > 1 , a ∈ L ∞ ( Ω ) , a ( ⋅ ) ≥ 0 , and 0 ν ≤ b ( ⋅ , ⋅ ) ≤ L . P is a double phase functional with mild transition between | D u | p and | D u | p log ⁡ ( e + | D u | ) . First, under suitable conditions on the moduli of continuity of a ( ⋅ ) and b ( ⋅ , ⋅ ) , we prove that local minimizers are of class C 0 , α for every α ∈ ( 0 , 1 ) , then that they are of class C 1 , α for some α > 0 , provided the functions a ( ⋅ ) and b ( ⋅ , ⋅ ) are Holder continuous.

中文翻译：

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具有温和过渡和正则系数的双相函数的极小值的正则性

摘要 我们证明了泛函 P ( w , Ω ) 的极小值的尖锐正则性结果： = ∫ Ω b ( x , w ) [ | Dw | p + a ( x ) | Dw | p log ⁡ ( e + | D w | ) ] dx , w ∈ W 1 , 1 ( Ω ) , p > 1 , a ∈ L ∞ ( Ω ) , a ( ⋅ ) ≥ 0 , 0 ν ≤ b ( ⋅ , ⋅ ) ≤ L 。P 是双相泛函，在 | 杜 | p 和 | 杜 | p log ⁡ ( e + | D u | ) 。首先，在 a ( ⋅ ) 和 b ( ⋅ , ⋅ ) 的连续模的合适条件下，我们证明局部极小值属于 C 0 类，对于每个 α ∈ ( 0 , 1 ) ，它们是类 C 1 , α 对于某些 α > 0 ，假设函数 a ( ⋅ ) 和 b ( ⋅ , ⋅ ) 是 Holder 连续的。