Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2402.09455 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910332172107776 |
|---|---|
| author | Yingxiao, Han Mi, Fang Liuye, Xia Hongya, Gao |
| author_facet | Yingxiao, Han Mi, Fang Liuye, Xia Hongya, Gao |
| contents | We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function $g$, and give applications. As a first application, we deal with variational integrals of the form $$ {\cal J} (u;Ω) = \int_Ω\ f(x,Du{(x)})dx. $$ We consider a minimizer $u: Ω\subset \mathbb R^n \to \mathbb R $ among all functions with a fixed boundary value $u_{\ast }$
on $\partial Ω$. Under some nonstandard growth conditions of the integrand $f(x,ξ)$ we derive some regularity results; as a second application, we consider elliptic equations of the form $$ \begin{cases} -\mbox {div} \left( a(x, u(x)) D u(x) \right) = f(x), & x \in Ω,
u(x) = 0, & x \in {\partial Ω}, \end{cases} $$ under the conditions $$ \frac {α}{(1+|s|) ^θ\ln ^θ(e+|s|)} \le a (x,s) \le β, \ \ \ 0<α\le β<\infty, \ θ\ge 0, $$ we obtain some regularity properties of its weak solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two Generalizations of Stampacchia Lemma and Applications Yingxiao, Han Mi, Fang Liuye, Xia Hongya, Gao Analysis of PDEs 35J20, 35J60 We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function $g$, and give applications. As a first application, we deal with variational integrals of the form $$ {\cal J} (u;Ω) = \int_Ω\ f(x,Du{(x)})dx. $$ We consider a minimizer $u: Ω\subset \mathbb R^n \to \mathbb R $ among all functions with a fixed boundary value $u_{\ast }$ on $\partial Ω$. Under some nonstandard growth conditions of the integrand $f(x,ξ)$ we derive some regularity results; as a second application, we consider elliptic equations of the form $$ \begin{cases} -\mbox {div} \left( a(x, u(x)) D u(x) \right) = f(x), & x \in Ω, u(x) = 0, & x \in {\partial Ω}, \end{cases} $$ under the conditions $$ \frac {α}{(1+|s|) ^θ\ln ^θ(e+|s|)} \le a (x,s) \le β, \ \ \ 0<α\le β<\infty, \ θ\ge 0, $$ we obtain some regularity properties of its weak solutions. |
| title | Two Generalizations of Stampacchia Lemma and Applications |
| topic | Analysis of PDEs 35J20, 35J60 |
| url | https://arxiv.org/abs/2402.09455 |