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We consider a stationary variational inequality with nonconstant gradient constraint and we prove the existence of solution of a Lagrange multiplier, assuming that the bounded open not necessarily convex set $\Omega$ has a smooth boundary.
If the gradient constraint $g$ is sufficiently smooth and satisfies $\Delta g^2\leq 0$ and the source term belongs to $L^\infty(\Omega)$, we are able to prove that the Lagrange multiplier belongs to $L^q(\Omega)$, for $1< q <\infty$, even in a very degenerate case. Fixing $q$, the result is still true if $\Delta g^2$ is bounded from above by a positive and sufficiently small number. Without the restriction on the signal of $\Delta g^2$ we obtain a weaker version of the above results.
This is a joint work with Lisa Santos.

We prove here the equivalence between many important properties concerning: the divergence operator, the Lions lemma, the Necas inequality, the Korn inequality and the weak lemma of Poincar\'e. Using then the Bogovskii operator and the Calderon-Zygmund theory, we give some isomorphism concerning the divergence operator. We give also a complete proof of the original De Rham theorem and we obtain some extension to the irrotational fields (see [1], [2]).
This is a joint work with Philippe G. Ciarlet and Cristinel Mardare.

References
[1] Amrouche Chérif, Ciarlet Philippe G., Mardare Cristinel. Remarks on a lemma by Jacques-Louis Lions, C. R. Math. Acad. Sci. Paris, 352-9, (2014), 691-695.
[2] Amrouche Chérif, Ciarlet Philippe G., Mardare Cristinel. On a lemma of Jacques-Louis Lions and its relation to other fundamental results, J. Math. Pures Appl. (9), 104, (2015), no. 2, 207-226.

In this talk, we focus on the numerical approximation of initial-boundary value problems to the diffusion equation with distributed-order in time. A brief review on the existing literature is provided and three different approaches are introduced, highlighting the main issues in the numerical approximation of such problems. Numerical examples for each of the approaches are presented and discussed.

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of degenerate elliptic type, which yields a precise control on the growth rate of solutions near their set of critical points. We then apply this new tool in the investigation of a longstanding conjecture which inquires whether solutions of the degenerate $p$-Poisson equation with a bounded source are locally of class $C^{1,1/p-1}$.
This is a joint work with Eduardo Teixeira and Damião Araújo.

Evento financiado pela FCT-Fundação para a Ciência e a Tecnologia, através do projeto UID/MAT/00013/2013.