Abstract: We formulate the flow of thick fluids as evolution quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor depending on the solution itself, and we show the existence of strong and weak solutions in viscous and inviscid cases. The results are based on new extensions of the continuous dependence of weak solutions to the variational inequalities for the Navier-Stokes and Euler equations with constraints on the derivatives, and on their respective generalised Lagrange multipliers.

This is a joint work with Lisa Santos.

Abstract: We discuss an inherently geometric argument, leading to new properties of convex functions defined on convex domains. Suppose the graph of (convex) Γ

 contains a line segment [Y^1,Y^2], where Y^j=(y^j,Γ(y^j)), with y^j∈Ω, and that Y∗=(y^∗,Γ(y^∗)) lies on this segment. Given a second-order polynomial P, whose graph touches (locally) the graph of Γ from below at Y^∗, then a horizontal/vertical translation of P/3 touches the convex graph of Γ at least at one of the points Y^j. This, in light of the viscosity approach in PDEs, has interesting consequences for the regularity of convex envelopes of supersolutions to a large class of partial differential equations. These include degenerate fully nonlinear models and quasi-linear problems that have not been treated in the literature.

This is joint work with Henrik Shahgholian (KTH, Sweden).

Abstract: We address the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term σ. When σ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this general setting. In the one-phase case, however, we obtain gradient bounds along the free boundary, revealing a structural gain in regularity. Most notably, we prove that if σ is continuous, then minimizers are of class C^1 along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.