The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911805503176704 |
|---|---|
| author | Ciani, Simone Henriques, Eurica Skrypnik, Igor i. |
| author_facet | Ciani, Simone Henriques, Eurica Skrypnik, Igor i. |
| contents | In this work we prove that the non-negative functions $u \in L^s_{loc}(Ω)$, for some $s>0$, belonging to the De Giorgi classes
\begin{equation}\label{eq0.1}
\fint\limits_{B_{r(1-σ)}(x_{0})} \big|\nabla \big(u-k\big)_{-}\big|^{p}\, dx \leqslant \frac{c}{σ^{q}} \,Λ\big(x_{0}, r, k\big)\bigg(\frac{k}{r}\bigg)^{p}\bigg(\frac{\big|B_{r}(x_{0})\cap\big\{u\leqslant k\big\}\big|}{|B_{r}(x_{0})|}\bigg)^{1-δ},
\end{equation}
under proper assumptions on $Λ$, satisfy a weak Harnack inequality with a constant depending on the $L^s$-norm of $u$. Under suitable assumptions on $Λ$, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth Ciani, Simone Henriques, Eurica Skrypnik, Igor i. Analysis of PDEs 35B40, 35B45, 35B65 In this work we prove that the non-negative functions $u \in L^s_{loc}(Ω)$, for some $s>0$, belonging to the De Giorgi classes \begin{equation}\label{eq0.1} \fint\limits_{B_{r(1-σ)}(x_{0})} \big|\nabla \big(u-k\big)_{-}\big|^{p}\, dx \leqslant \frac{c}{σ^{q}} \,Λ\big(x_{0}, r, k\big)\bigg(\frac{k}{r}\bigg)^{p}\bigg(\frac{\big|B_{r}(x_{0})\cap\big\{u\leqslant k\big\}\big|}{|B_{r}(x_{0})|}\bigg)^{1-δ}, \end{equation} under proper assumptions on $Λ$, satisfy a weak Harnack inequality with a constant depending on the $L^s$-norm of $u$. Under suitable assumptions on $Λ$, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents. |
| title | The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth |
| topic | Analysis of PDEs 35B40, 35B45, 35B65 |
| url | https://arxiv.org/abs/2403.13539 |