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Weighted gradient inequalities and unique continuation problems
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-04-22 , DOI: 10.1007/s00526-020-1716-8 Laura De Carli , Dmitry Gorbachev , Sergey Tikhonov
中文翻译:
加权梯度不等式和唯一的连续问题
更新日期:2020-04-23
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-04-22 , DOI: 10.1007/s00526-020-1716-8 Laura De Carli , Dmitry Gorbachev , Sergey Tikhonov
We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality
$$\begin{aligned} \Vert e^{-\tau \ell (\cdot )} u^{\frac{1}{q}} f\Vert _q\le c_\tau \Vert e^{-\tau \ell (\cdot )} v^{\frac{1}{p}}\, \nabla f\Vert _p, \quad f\in C^\infty _0({\mathbb {R}}^n). \end{aligned}$$This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.
中文翻译:
加权梯度不等式和唯一的连续问题
我们将Pitt不等式用于Fourier变换以证明以下加权梯度不等式
$$ \ begin {aligned} \ Vert e ^ {-\ tau \ ell(\ cdot)} u ^ {\ frac {1} {q}} f \ Vert _q \ le c_ \ tau \ Vert e ^ {-\ tau \ ell(\ cdot)} v ^ {\ frac {1} {p}} \,\ nabla f \ Vert _p,\ quad f \ in C ^ \ infty _0({\ mathbb {R}} ^ n) 。\ end {aligned} $$该不等式是Carleman型估计,对于一阶微分方程和系统的解产生唯一的连续结果。