Évènements

Existence and boundedness of solutions to singular anisotropic elliptic equations

Catégorie d'évènement : Séminaire Équations aux Derivées Partielles et Applications (Nancy) Date/heure : 16 avril 2024 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Florica Cirstea (Université de Sydney) Résumé :
In this talk, we present new results on the existence and uniform boundedness of solutions for a general class of Dirichlet anisotropic elliptic problems
of the form
$$ -\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f \quad \mbox{in } \Omega, \qquad u=0 \quad \mbox{on }\partial \Omega,$$
where $\Omega$ is a bounded domain in $ \mathbb R^N$ $(N\geq 2)$, $ \Delta_{\overrightarrow{p}}u=\sum_{j=1}^N \partial_j (|\partial_j u|^{p_j-2}\partial_j u)$ and
$\Phi_0(u,\nabla u)=\left(\mathfrak{a}_0+\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}\right)|u|^{m-2}u$,
with $\mathfrak{a}_0>0$,
$m,p_j>1$,   $\mathfrak{a}_j\geq 0$ for $1\leq j\leq N$ and $N/p=\sum_{k=1}^N (1/p_k)>1$. We assume that $f \in L^r(\Omega)$ with $r>N/p$. The feature of this study  is the inclusion of a possibly singular gradient-dependent term $\Psi(u,\nabla u)=\sum_{j=1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j}$, where $\theta_j>0$ and $0\leq q_j<p_j$ for $1\leq j\leq N$.
This is joint work with Barbara Brandolini (Università degli Studi di Palermo).

Colloquium : Felix Otto (Max Planck Institute for Mathematics in the Sciences, Leipzig)

Catégorie d'évènement : Colloquium Date/heure : 16 avril 2024 16:30-17:30 Lieu : Salle de conférences Nancy Oratrice ou orateur : Felix Otto (Max Planck Institute for Mathematics in the Sciences, Leipzig) Résumé :

Titre : Singular stochastic partial differential equations: more geometry and less combinatorics

Résumé :

Singular stochastic partial differential equations are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. The guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow the approach of mathematical physics, and of Hairer’s regularity structures, which however we re-interpret as providing an informal parameterization of the infinite-dimensional nonlinear solution manifold.

We systematically follow this more geometric/analytic than combinatorial point-of-view: Instead of appealing to an expansion indexed by trees, we consider all partial derivatives w. r. t. the « constitutive » function defining the nonlinearity. Instead of a Gaussian calculus guided by Feynman diagrams arising from pairing nodes of two trees, we consider derivatives w. r. t. the noise, i. e. Malliavin derivatives. We interpret the Malliavin derivative of the parameterization as an approximate tangent vector to the solution manifold, which yields a sparse representation in terms of the parameterization itself, and paves the way for its stochastic estimate. Ultimately, this gives a characterization of the solution manifold that is oblivious to the divergent counter terms.

This is work with L. Broux, P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.