# Mathematics Department Colloquium Series

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The Mathematics Department established the Mathematics Department Colloquium Series through a Faculty Development Grant in 1999 to provide a place to discuss new research in Mathematics. Speakers for the series come from both inside the UTC Department of Mathematics as well as guest speakers from other institutions. The Colloquium Series meets periodically throughout the year.

### Most Recent Colloquium Series Talk

**Lingju Kong **

Thursday, October 3, EMCS 211, 4:25-5:15 pm.

**On a Fourth Order Elliptic Problem with a p(x)-Biharmonic Operator**

**Abstract**. Differential equations and variational problems with nonstandard p(x) -growth conditions
have many applications in mathematical physics such as in the modelling of electrorheological
fluids and of other phenomena related to image processing, elasticity, and the flow
in porous media. In this work, we study the fourth order nonlinear eigenvalue problem
with a p(x) -biharmonic operator

\Delta^2_{p(x)u+a(x)|u|^{p(x)-2u=\lambda w(x)f(u)\quad \rm{in\ \Omega, u=\Delta u=0\quad \rm{on\ \partial\Omega,

where \Omega is a smooth bounded domain in R^N, p\in C(\overline{\Omega) with p(x)>1 on \overline{\Omega}, \Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the p(x) -biharmonic operator, and \lambda>0 is a parameter. Under some appropriate conditions on the functions p, a, w, f, we prove that there exist \overline{\lambda}>0 and \underline{\lambda}>0 such that any \lambda \in (0,\overline{\lambda}) and \lambda \in (\underline{\lambda}, \infty) is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments in critical point theory and some recent theory on the generalized Lebesgue--Sobolev spaces L^{p(x)(\Omega) and W^{k,p(x)}(\Omega).

*** This talk will be appropriate for graduate students with an interest in Differential
Equations.*

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