Ước Lượng Độ Dốc Lorentz Có Trọng Số cho Một Lớp Phương Trình Elliptic Quasilinear với Dữ Liệu Đo Đạc

Bulletin of the Iranian Mathematical Society - 2020
Fengping Yao1
1Department of Mathematics, Shanghai University, Shanghai, China

Tóm tắt

Trong bài báo này, chúng tôi thu được các ước lượng độ dốc Lorentz có trọng số cục bộ như sau: $$\begin{aligned} g^{-1}\left( {\mathcal {M}}_1(\mu ) \right) \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \Longrightarrow |Du| \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \end{aligned}$$ cho các nghiệm yếu của một lớp các phương trình elliptic quasilinear không đồng nhất với dữ liệu đo đạc $$\begin{aligned} -\text {div} ~\! \left( a\left( \left| \nabla u \right| \right) \nabla u \right) = \mu , \end{aligned}$$ trong đó $$g(t)= t a(t)$$ với $$t\ge 0$$ và $$\begin{aligned} {\mathcal {M}}_1(\mu )(x):=\sup _{r>0}\frac{r|\mu |(B_r(x))}{|B_r(x)|}, \quad x\in {\mathbb {R}}^{n}. \end{aligned}$$ Hơn nữa, chúng tôi nhận thấy rằng hai ví dụ tự nhiên và đơn giản của các hàm g(t) trong công trình này là $$\begin{aligned} g(t)=t^{p-1} ~~(p\text{-Laplace } \text{ phương trình) }~~~~~~ \text{ và } ~~~~~~ g(t)=t^{p-1 }\log ^\alpha \big ( 1+t\big ) \quad \text{ cho }~\alpha > 0. \end{aligned}$$ Thực tế, ví dụ tổng quát và thú vị hơn có liên quan đến điều kiện tăng trưởng (p, q) bằng cách ghép nối thích hợp các đa thức. Chúng tôi nhận thấy rằng các kết quả của chúng tôi cải thiện các kết quả đã biết cho các phương trình như vậy.

Từ khóa

#Gradient Lorentz #phương trình elliptic quasilinear #dữ liệu đo đạc #ước lượng trọng số

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Bulletin of the Iranian Mathematical Society

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