论文标题
传输问题的最佳规律性] {自由传输问题的最佳规律性
Optimal Regularity in Transmission Problems]{Optimal regularity for variational solutions of free transmission problems
论文作者
论文摘要
在本文中，我们研究了\ cite {hs21}中考虑的类型的功能，即$$ j（v）：= \ int_ {b_1} a（x，x，x，u）| \ nabla u |^2 + f（x，x，x，x，x，u + + + + + + + + q（x）λ（x）λ（u）λ（u）λ（u）λ\，dx $ $ a _+（x）χ_ {\ {U> 0 \}}+a _-（x）χ_ {\ {u <0 \}} $，$ f（x，x，u）= f _+（x）= f _+（x）χ_ { $λ（x，u）=λ_ +（x）χ_ {\ {u> 0 \}}} +λ_-（x）χ_ {\ \ {U \ {u \ le 0 \}} $。我们证明了上述功能的最低量的最佳$ c^{0,1^ - } $（具有精确的Hölder估算值）时，当系数$ a _ _ {\ pm} $是连续函数和$μ\ le a _ _ _ _ {\ pm} {\ pm} \ le \ le \ freac $ 0 $ 0 $ 0 $ 0 $ 0 $ 0 $ 0 $ 0 <1 l^n（b_1）$和$ q $界。我们通过提出一种新的紧凑性论证和近似理论来做到这一点，类似于caffarelli在\ cite {Ca89}中开发的一种，以处理完全非线性PDE的解决方案的规则性理论。此外，我们介绍了$ \ MATHCAL {T} _ {a，b} $操作员，该操作员允许人们将最小化器从传输问题传递到Alt-Caffarelli-Friedman类型函数，{以小规模，}以这种方式研究Bernoulli Type Type Type Type Type Type Type Type Typel Typel Thress Type Type Type Type Type Trapel Throxpress of Type Type Typel Thrapel Thress Type Type Type Type Typel Typel Thraxply问题的方法。
In this article we study functionals of the type considered in \cite{HS21}, i.e. $$ J(v):=\int_{B_1} A(x,u)|\nabla u|^2 +f(x,u)u+ Q(x)λ(u)\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u<0\}}$, $f(x,u)= f_+(x)χ_{\{u>0\}}+f_-(x) χ_{\{u<0\}}$ and $λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove the optimal $C^{0,1^-}$ regularity of minimizers of the functional indicated above (with precise Hölder estimates) when the coefficients $A_{\pm}$ are continuous functions and $μ\le A_{\pm}\le \frac{1}μ$ for some $0<μ<1$, with $f \in L^N(B_1)$ and $Q$ bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in \cite{Ca89} to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the $\mathcal{T}_{a,b}$ operator that allows one to transfer minimizers from the transmission problems to the Alt-Caffarelli-Friedman type functionals, {in small scales,} allowing this way the study of the regularity theory of minimizers of Bernoulli type free transmission problems.
