Speaker: Akshara Vincent (University of Pittsburgh)

Title: Convex integration techniques for non-uniqueness and failure of Calderón-Zygmund estimates

Abstract: In this talk, we discuss the application of convex integration to the construction of irregular solutions to nonlinear partial differential equations in certain classes of Sobolev spaces. We will introduce the recent works of Colombo and Tione (2022), where they proved the non-uniqueness of solutions of the p-Laplace equation in class $W^{1,r}_0$ for r<2, and of Schikorra (2024) on the failure of corresponding Calderón-Zygmund estimates. Inspired from their works, we study the following nonlinear problem: for a given domain $Omegasubset mathbb{R}^2$ and a given mapping $A:mathbb{R}^2to mathbb{R}^2$, to study the solutions $u:Omegato mathbb{R}$ satisfying

begin{equation*}

begin{cases}

operatorname{div}(A(nabla u))=0 &text{in }Omega

u=0 &text{on }partial Omega.

end{cases}

end{equation*}

We will see an example of a function A which is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck, which exhibits non-uniqueness of solutions and failure of a priori Calderón-Zygmund estimates below a critical exponent.  

Zoom link: https://wayne-edu.zoom.us/j/98693739093?pwd=wrzcgvyxMFfqar3ognCZjIpk4UsSfi.1