Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data
In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces $$ g^{-1} \left[\mathcal M_1(\mu) \right] \in L^{q, r}_{w, loc} (\Omega) \Longrightarrow |Du| \in L^{q, r}_{w, loc} (\Omega), $$ where $g(t)= t a(t)$ for $t\geq 0$ and $\mathcal{M}_1(\mu)(x)$ is...
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University of Szeged
2024-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10788 |
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| _version_ | 1841533466517176320 |
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| author | Fengping Yao |
| author_facet | Fengping Yao |
| author_sort | Fengping Yao |
| collection | DOAJ |
| description | In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces
$$
g^{-1} \left[\mathcal M_1(\mu) \right] \in L^{q, r}_{w, loc} (\Omega) \Longrightarrow |Du| \in L^{q, r}_{w, loc} (\Omega),
$$
where $g(t)= t a(t)$ for $t\geq 0$ and $\mathcal{M}_1(\mu)(x)$ is the first-order fractional maximal function
$$
\mathcal{M}_1(\mu)(x):=\sup_{r>0}\frac{r|\mu|(B_r(x))}{|B_r(x)|},
$$
for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case
\begin{equation*}
-\operatorname{div}\left[a \left( \left( A D u \cdot D u\right)^{\frac{1}{2}} \right)A D u \right]
=\mu \quad \mbox{in}~ \Omega,
\end{equation*}
whose model cases are the classical elliptic $p$-Laplacian equation with measure data
\begin{align*}
-\operatorname{div} \left( \left| D u \right|^{p-2}
D u \right)=\mu \quad \mbox{for } 1<p<2
\end{align*}
and the elliptic $p$-Laplacian equation with the logarithmic term and measure data
\begin{equation*}
-\operatorname{div} \left( \left| D u \right|^{p-2}\log \left(1+ \left| D u \right|\right)
D u \right)=\mu \quad \mbox{for } 1<p<2.
\end{equation*}
It deserves to be specially noted that the subquadratic case is a little different from the superquadratic case since as an example, the modulus of ellipticity in the above-mentioned special cases tends to infinity when $|Du| \rightarrow 0$ for $1<p<2$. |
| format | Article |
| id | doaj-art-1b7daf4599a240839a51343f2ac09f52 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-1b7daf4599a240839a51343f2ac09f522025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-01-012024712510.14232/ejqtde.2024.1.710788Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure dataFengping Yao0Department of Mathematics, Shanghai University, Shanghai, China; Newtouch Center for Mathematics of Shanghai University, Shanghai, ChinaIn this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces $$ g^{-1} \left[\mathcal M_1(\mu) \right] \in L^{q, r}_{w, loc} (\Omega) \Longrightarrow |Du| \in L^{q, r}_{w, loc} (\Omega), $$ where $g(t)= t a(t)$ for $t\geq 0$ and $\mathcal{M}_1(\mu)(x)$ is the first-order fractional maximal function $$ \mathcal{M}_1(\mu)(x):=\sup_{r>0}\frac{r|\mu|(B_r(x))}{|B_r(x)|}, $$ for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case \begin{equation*} -\operatorname{div}\left[a \left( \left( A D u \cdot D u\right)^{\frac{1}{2}} \right)A D u \right] =\mu \quad \mbox{in}~ \Omega, \end{equation*} whose model cases are the classical elliptic $p$-Laplacian equation with measure data \begin{align*} -\operatorname{div} \left( \left| D u \right|^{p-2} D u \right)=\mu \quad \mbox{for } 1<p<2 \end{align*} and the elliptic $p$-Laplacian equation with the logarithmic term and measure data \begin{equation*} -\operatorname{div} \left( \left| D u \right|^{p-2}\log \left(1+ \left| D u \right|\right) D u \right)=\mu \quad \mbox{for } 1<p<2. \end{equation*} It deserves to be specially noted that the subquadratic case is a little different from the superquadratic case since as an example, the modulus of ellipticity in the above-mentioned special cases tends to infinity when $|Du| \rightarrow 0$ for $1<p<2$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10788weightedlorentzgradientsubquadraticquasilinear ellipticmeasure |
| spellingShingle | Fengping Yao Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data Electronic Journal of Qualitative Theory of Differential Equations weighted lorentz gradient subquadratic quasilinear elliptic measure |
| title | Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| title_full | Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| title_fullStr | Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| title_full_unstemmed | Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| title_short | Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| title_sort | weighted lorentz estimates for subquadratic quasilinear elliptic equations with measure data |
| topic | weighted lorentz gradient subquadratic quasilinear elliptic measure |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10788 |
| work_keys_str_mv | AT fengpingyao weightedlorentzestimatesforsubquadraticquasilinearellipticequationswithmeasuredata |