A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010). These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.Comment: 58 pages, 2 figure

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category \Tops of stratified spaces, that are topological spaces $X$ endowed with a partition \cF and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,\cF) of \Tops together with a class \cA of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If (X,\cF)\in\Tops, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any S\in \cF. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with \cA the class of transverse subsets, coincide with the tangential category and are homotopical invariants.Comment: 14 pages, 2 figure

Deep ductile shear localization facilitates near-orthogonal strike-slip faulting in a thin brittle lithosphere

Some active fault systems comprise near-orthogonal conjugate strike-slip faults, as highlighted by the 2019 Ridgecrest and the 2012 Indian Ocean earthquake sequences. In conventional failure theory, orthogonal faulting requires a pressure-insensitive rock strength, which is unlikely in the brittle lithosphere. Here, we conduct 3D numerical simulations to test the hypothesis that near-orthogonal faults can form by inheriting the geometry of deep ductile shear bands. Shear bands nucleated in the deep ductile layer, a pressure-insensitive material, form at 45 degree from the maximum principal stress. As they grow upwards into the brittle layer, they progressively rotate towards the preferred brittle faulting angle, ~30 degree, forming helical shaped faults. If the brittle layer is sufficiently thin, the rotation is incomplete and the near-orthogonal geometry is preserved at the surface. The preservation is further facilitated by a lower confining pressure in the shallow portion of the brittle layer. For this inheritance to be effective, a thick ductile fault root beneath the brittle layer is necessary. The model offers a possible explanation for orthogonal faulting in Ridgecrest, Salton Trough, and Wharton basin. Conversely, faults nucleated within the brittle layer form at the optimal angle for brittle faulting and can cut deep into the ductile layer before rotating to 45 degree. Our results thus reveal the significant interactions between the structure of faults in the brittle upper lithosphere and their deep ductile roots