The classical Friedrichs identity states that for $u\in C_0^{\infty}(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n} |D^2 u|^2\, dx = \int_{\mathbb{R}^n} |\Delta u|^2\, dx.$$
From this inequality we immediately get $W^{2,2}$-estimates for
solutions of $\Delta u =f$ and also for solutions of measurable perturbations
of the form $\sum_{ij}a_{ij}(x) u_{ij}(x)=f(x)$, when the matrix
$A=(a_{ij})$ is closed to the identity in sense made precise
by Cordes.
In this talk we first explore extensions of the Friedrichs identity in
the form of sharp inequalities
$$\int_{X} |\mathfrak{X}^2 u|^2\, dx \le C_1 \int_{X} |\Delta_{\mathfrak{X}} u|^2\, dx +C_2\int_{X} |\mathfrak{X}u|^2 \, dx $$
where $X$ is a Riemannian manifold, the Heisenberg group, and certain types of CR manifolds.
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We then show how to use these estimates to study quasilinear subelliptic equations.\par

This is joint work with Andr\`as Domokos (PAMS 133, 2005) and Sagun
Chanillo (2007 preprint.)

Diffusion, along with transport and reaction effects, is the main factor explaining changes
or transitions in a wide array of situations such as flames, some phase transitions, tumours
or other biological invasions. In these systems, two or several possible states coexist, and
one observes certain states expanding or receding or patterns being formed.

This lecture, meant for a general audience, will describe some mathematical properties of
reaction-diffusion equations as an approach to spatial propagation and diffusion. After
describing the mechanism of reaction and diffusion and giving several illustrations, I will
review some classical results. In the context of ecology of populations, I will then mention
some recent works dealing with non homogeneous media. In this framework, I will describe a
model addressing the question of how a species keeps pace with a shifting climate.

The interaction of flowing fluids with free bodies -- sometimes
compliant, sometimes active, sometimes multiple -- constitutes a class
of beautiful dynamic boundary problems that are central to biology and
engineering. Examples range from how organisms locomote in fluids
(which depends strongly on scale) to how non-Newtonian stresses
develop in complex liquids (strongly dependent on the nature of
fluidic microstructure). I will discuss several interesting examples,
emphasizing how they are formulated mathematically so as to yield
models tractable for analysis or simulation, and show how this work
has interacted with experimental studies.