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### 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

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 $B=F_s$.