We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min⁡{∫Ωf⁢(x,D⁢v⁢(x)):v∈Kψ⁢(Ω)},\min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1<p<21<p<2, and Kψ⁢(Ω)\mathcal{K}_{\psi}(\Omega) is the class of admissible functions v∈u0+W01,p⁢(Ω)v\in u_{0}+W^{1,p}_{0}(\Omega) such that v≥ψv\geq\psi a.e. in Ω, where u0∈W1,p⁢(Ω)u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x↦Dξ⁢f⁢(x,ξ)x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f⁢(x,ξ)≈a⁢(x)⁢|ξ|pf(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1<p<21<p<2, and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

中文翻译：

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具有次二次增长条件的一类非自治障碍问题的解的更高可微性结果

对于形式为min⁡{∫Ωf⁢（x，D⁢v⁢（x））：v∈Kψ⁢（Ω）}，\的非自治障碍问题，我们建立了一些整数阶和分数阶更高的微分结果。 min \ biggl {\ {} \ int _ {\ Omega} f（x，Dv（x））：v \ in \ mathcal {K} _ {\ psi}（\ Omega）\ biggr {\}}，其中函数对于1 <p <21 <p <2，满足梯度变量的𝑝-增长条件，并且Kψ⁢（Ω）\ mathcal {K} _ {\ psi}（\ Omega）是可允许函数的类别v∈u0+ W01，p⁢（Ω）v \ in u_ {0} + W ^ {1，p} _ {0}（\ Omega）使得v≥ψv\ geq \ psi ae inΩ，其中u0∈ W ^ {1，p}（\ Omega）中的W1，p⁢（Ω）u_ {0} \是固定的边界基准。在这里，我们显示了障碍物the的坡度的Sobolev或Besov–Lipschitz正则性假设转移到了解决方案的坡度，前提是提供了局部映射xmapDξ⁢f⁢（x，ξ）x \ mapsto D _ {\ xi} f（x，\ xi）属于合适的Sobolev或Besov空间。