## View abstract

### Session S18 - Recent progress in non-linear PDEs and their applications

Monday, July 12, 14:00 ~ 14:50 UTC-3

## Lipschitz continuity of nonnegative minimizers of functional of Bernoulli type with nonstandard growth

### Noemi Wolanski

#### IMAS-UBA-CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak73bd86a64415cd9e8c6ef9ec3eb1f1f6').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy73bd86a64415cd9e8c6ef9ec3eb1f1f6 = 'w&#111;l&#97;nsk&#105;' + '&#64;'; addy73bd86a64415cd9e8c6ef9ec3eb1f1f6 = addy73bd86a64415cd9e8c6ef9ec3eb1f1f6 + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_text73bd86a64415cd9e8c6ef9ec3eb1f1f6 = 'w&#111;l&#97;nsk&#105;' + '&#64;' + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloak73bd86a64415cd9e8c6ef9ec3eb1f1f6').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy73bd86a64415cd9e8c6ef9ec3eb1f1f6 + '\'>'+addy_text73bd86a64415cd9e8c6ef9ec3eb1f1f6+'<\/a>';

I will report on research done in collaboration with Claudia Lederman from the University of Buenos Aires on the Lipschitz continuity of nonnegative minimizers to functionals $J(u)=\int_\Omega F(x,u(x),\nabla u(x))+\lambda(x)\chi_{\{u>0\}}\,dx.$

Here $F(x,s,\eta)$ is a function of $p(x)-$type growth with $p$ Lipschitz continuous, and $0<\lambda_1\le \lambda(x)\le\lambda_2<\infty$.

Some examples are $F(x,s,\eta)=a(x,s)|\eta|^{p(x)}+b(x) |s|^{p^*(x)}$, or $F(x,s,\eta)=G(|\eta|^{p(x)})+b(x) |s|^{p^*(x)}$ with $G$ strictly convex, under suitable assumptions.

Of independent interest are the results on existence, $L^\infty-$estimates, comparison principles and maximum principles for the associated equation $\mbox{div}\big(A(x,u(x),\nabla u(x))\big)=B(x,u(x),\nabla u(x)),$ where $A=\nabla _\eta F$ and $B=F_s$.

Joint work with Claudia Lederman (Universidad de Buenos Aires, Argentina).