Pavia (Italy), September 24-25, 2021

Titles and abstracts

Invited Lectures

Invited Lectures

  •  V. Boegelein: Higher regularity in congested traffic dynamics

    Abstract: We consider an elliptic system that is motivated by a congested traffic dynamics problem. It has the form $$ \mathrm{div}\bigg((|Du|-1)_+^{p-1}\frac{Du}{|Du|}\bigg)=f, $$ and falls into the context of very degenerate problems. Continuity properties of the gradient have been investigated in the scalar case by Santambrogio & Vespri and Colombo & Figalli. In this talk we establish the optimal regularity of weak solutions in the vectorial case for any $p$>$1$. This is joint work with F. Duzaar, R. Giova and A. Passarelli di Napoli.

  •  M. Bonforte: Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

    Abstract: We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m\in\big((N-2)_+/(N+2),1\big)$. It is known that bounded positive solutions extinguish in a finite time $T>0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$, where $S$ belongs to the set of solutions to a suitable elliptic problem and depends on the initial datum $u_0$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincaré inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of ``almost orthogonality'', which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincar\'e inequalities and sharp convergence rates for linear flows. This is a joint work with Alessio Figalli (ETH Zürich).

  •  G. Da Prato: A New Probabilistic Formula for the Gradient of Solutions of Some Hypoelliptic Dirichlet Problems

    Abstract: We are concerned with the following Cauchy-Dirichlet problem, \begin{equation}\label{eq} \left\lbrace \begin{aligned} &D_t u(t,x)=\tfrac12\text{Tr}[CD^2_x u(t,x)]+\langle Ax,D_x u(t,x)\rangle,\quad (t,x)\in{\mathbb R}^+\times\overline{{\mathcal O}},\\ &u(t,x)=0,\quad t\ge0,\ x\in\partial{\mathcal O},\\ &u(0,x)=\varphi(x),\quad x\in\overline{{\mathcal O}}. \end{aligned} \right. \end{equation} $A$ and $C$ are $d\times d$ matrices with $C$ semidefinite positive and singular and ${\mathcal O}$ is a bounded convex open set of ${\mathbb R}^d$. Our basic assumption is the following

    Hypothesis [Hypoellipticity]
    The matrix $Q_t:=\int_0^t e^{sA} C e^{sA^*}ds$ is non singular for all $t>0$.

    The formal solution $u(t,x)$ of the Cauchy-Dirichlet problem is given by $R^{\mathcal O}_T\varphi(x)$ where $R^{\mathcal O}_T\varphi$ is the stopped semigroup $$ R^{\mathcal O}_T\varphi(x)={\mathbb E}[\varphi(X(T,x)){\mathbb 1}_{T\le\tau_x}],\ T>0,\quad x\in\overline{{\mathcal O}} $$ where $$ X(T,x)=e^{tA}x+\int_0^Te^{(t-s)A}dW(s) $$ and $W(t)$, $t\ge0$, is an ${\mathbb R}^d$-valued standard Wiener process defined on a probability space $(\Omega,{\mathcal F},{\mathbb P})$.
    Moreover, $\tau_x$ is the exit time from $\overline{{\mathcal O}}$, $$ \tau_x=\inf\{t\ge0:\ X(t,x)\in\overline{{\mathcal O}}^c\}. $$ We prove a new representation formula of the gradient of $R^{\mathcal O}_T\varphi(x)$, for all $T>0$, $x\in\overline{{\mathcal O}}$ and all $\varphi$ bounded and Borel on $\overline{{\mathcal O}}$.
    This is a joint work with Luciano Tubaro.

  •  F. Duzaar: Boundary regularity for elliptic systems with $p$,$q$-growth

    Abstract: We investigate the boundary regularity of minimizers of convex integral functionals with nonstandard $p$,$q$-growth and with Uhlenbeck structure. We consider arbitrary convex domains $\Omega$ and homogeneous Dirichlet data on some part $\Gamma\subset\partial\Omega$ of the boundary. For the integrand we assume only a non-standard $p$,$q$-growth condition. We establish Lipschitz regularity of minimizers up to $\Gamma$, provided the gap between the growth exponents $p$ and $q$ is not too large, more precisely if $1$<$p\le q$<$p\left(1+\frac2n\right)$. To our knowledge, this is the first boundary regularity result under a non-standard $p$,$q$-growth condition case for any $p$>$1$.
    This is joint work with Verena Bögelein, Paolo Marcellini and Christoph Scheven.

  •  J. Kinnunen: Supercaloric functions for the parabolic $p$-Laplace equation in the fast diffusion case

    Abstract: This talk discusses a generalized class of supersolutions, so-called $p$-supercaloric functions, to the parabolic $p$-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood in the slow diffusion case $p$>$2$, but little is known in the fast diffusion case $1$<$p$<$2$. For $p=2$ we have supercaloric functions for the heat equation.
    Every bounded $p$-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic $p$-Laplace equation for the entire range $1$<$p$<$\infty$. In the slow diffusion case every unbounded $p$-supercaloric function has either a Barenblatt type behavior or blows up at least with the rate given by the friendly giant. Our main result is the corresponding result in the supercritical case $\frac{2n}{n+1}$<$p$<$2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur and that the obtained estimates are sharp. The theory is not yet well understood in the subcritical case $1$<$p\le \frac{2n}{n+1}$.

  •  E. Lanconelli: The caloric Dirichlet problem: Perron solution, Wiener solution, Lebesgue solution

    Abstract: We compare the Perron, Wiener and Lebesgue methods to construct generalized solution of the Dirichlet problem for the heat equation on general bounded open subset of the Euclidean space-time. We also show how to construct the Perron solution with a very elementary procedure.

  •  J. J. Manfredi: Asymptotic Mean Value Expansions for Solutions of General Elliptic and Parabolic Equations

    Abstract: We obtain asymptotic mean value formulas for solutions of second-order elliptic and parabolic equations. Our approach is flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). Examples include Pucci, Issacs, and k-Hessian operators and some of their parabolic versions.
    This talk is based in joint work with Pablo Blanc (Jyväskylä), Fernando Charro (Detroit), and Julio Rossi (Buenos Aires).

  •  J. M. Urbano: Sharp regularity for singular and degenerate pdes

    Abstract: We provide a broad overview of qualitative versus quantitative regularity estimates in the theory of singular and degenerate parabolic pdes. The former relate to the method of intrinsic scaling, while the latter are achieved by means of geometric tangential analysis. We discuss, in particular, sharp estimates for the Stefan problem, the parabolic $p$-Poisson equation, the porous medium equation and Trudinger's equation.

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