## Abstracts

 Associate Professor @ University of Sydney, Australia. A/Professor Florica Cirstea graduated with a PhD from Victoria University in April 2005. From April 2005 to April 2009, she held an ARC Australian Postdoctoral Fellowship (APD). The teaching (25%) was funded by the Department of Mathematics at the Australian National University and the research (75%) was funded by the Australian Research Council. In July 2008, she transferred her fellowship to the University of Sydney to take up a lectureship position in April 2009. She is an Associate Professor at the University of Sydney since January 2017. Anisotropic elliptic equations with gradient-dependent lower order terms and $$L^1$$ data Given a bounded open subset $$\Omega$$ of $$\mathbb R^N$$, we prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems including $$\mathcal Au+\Phi(x,u,\nabla u)=\mathfrak{B}u+f$$ in $$\Omega$$, where $$f\in L^1(\Omega)$$ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $$\mathcal A$$, the prototype of which is $$\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u)$$ with $$p_j>1$$ for all $$1\leq j\leq N$$ and $$\sum_{j=1}^N (1/p_j)>1$$. As a novelty, our lower order terms involve a new class of operators $$\mathfrak B$$ such that $$\mathcal{A}-\mathfrak{B}$$ is bounded, coercive and pseudo-monotone from $$W_0^{1,\overrightarrow{p}}(\Omega)$$ into its dual, as well as a gradient-dependent nonlinearity $$\Phi$$ with an “anisotropic natural growth" in the gradient and a good sign condition. This talk is based on joint work with Barbara Brandolini (Università degli Studi di Palermo).

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