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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-01-01
|
| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10788 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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 $1<p<2$. |
|---|---|
| ISSN: | 1417-3875 |