Lisbon WADE — Webinar in Analysis and Differential Equations

Recent developments on Dirichlet problems with singular convection/drift terms

In the talk we discuss two Dirichlet problems ("formally" in duality)\begin{align}\label{(CP)}\tag{CP}u\in W_0^{1,q}(\Omega): \; &\operatorname{div}(M(x)\nabla {u})+a(x)\,{u}=-\operatorname{div}({u}\,E(x)) +f(x);
\\ \label{(DP)}\tag{DP}\psi\in W_0^{1,q}(\Omega): & - \operatorname{div}(M(x)\nabla \psi)+a(x)\,\psi= E(x)\cdot\nabla \psi +g(x)\end{align}where $\Omega$ is a bounded open set in $\mathbb{R}^N$, $M(x)$ ia bounded elliptic matrix, $f$, $g$ are functions belonging to $L^m(\Omega)$, $m\geq1$, $E\in(L^N(\Omega))^N$, $0\lt \alpha_0\leq a(x)\in L^1(\Omega)$.

In the first part we briefly recall some recent results:

• existence and summability properties of weak solutions ($q=2$), if $m\geq\frac{2N}{ N+1}$;
• Calderon-Zygmund theory ($q=m^*$, infinite energy solutions), if $1\lt m \lt \frac{2N}{ N+1}$;
• uniqueness;
• the case $|E|\leq\frac{A}{|x|}$, where $E\not\in(L^N(\Omega))^N$;
• the case $E\in(L^2(\Omega))^N$.

Then we show:

• a new (simpler) existence proof, thanks to the presence of the zero order term, for \eqref{(CP)};
• a straight duality proof for \eqref{(DP)};
• continuous dependence of the solutions with respect to the weak convergence of the coefficients;
• regularizing effect of dominated coefficients ($|f|\leq Q\,a(x)$ or $|g|\leq Q\,a(x)$, $Q\gt 0$);
• "weak" maximum principle: if $f(x)\geq0$ [$g(x)\geq0$] and, of course, not =0 a.e., then $u(x)\geq0$ [$\psi(x)\geq0$] and the set where $u$ [$\psi$] is zero has zero measure (at most).

Work in progress: obstacle problem; nonlinear principal part.
Open problem: "strong" maximum principle.

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Organizers: Hugo Tavares, James Kennedy and Nicolas Van Goethem

Joint iniciative of the research centers CAMGSD, CMAFcIO and GFM.