Calculus of Variations and Geometric Measure Theory

Convex functions in the Heisenberg Group

Bianca Stroffolini (Dip. Mat. Univ. Napoli)

created by paolini on 14 Feb 2002

28 feb 2002


Prof. Bianca STROFFOLINI (Universita' di Napoli) ``Convex functions in the Heisenberg Group'' Dipartimento di Matematica - Sala dei Seminari Gioved\`\i\ 28 Febbraio - ore 17.00


Regularizing properties of inf-convolutions were described in Lasry-Lions and used by Jensen to prove his celebrated comparison principle for second order fully nonlinear elliptic equations. Shortly after Jensen-Lions-Souganidis suggested to look at the time-dependent eikonal equation: $$\frac{\partial w}{\partial t}+\frac1{2}
{2}=0 ,$$ $$\frac{\partial w}{\partial t}-\frac1{2}
{2}=0 ,$$ with initial data $w(\cdot,0)=u$ or $w(\cdot,0)=v$, $u$ upper semicontinuous and $v$ lower semicontinuous. The solutions of these two equations are given by explicit formulas in terms of sup and inf convolution respectively (Hopf-Lax formulas): $$u{\varepsilon}(x)=\sup{y} \left\{u(y)-\frac1{2\varepsilon}
{2} \right\}$$ $$v{\varepsilon}(x)=\inf{y} \left\{v(y)+\frac1{2\varepsilon}
{2} \right\}$$

They enjoy good semiconvexity and semiconcavity properties respectively and they preserve the subsolution and supersolution character of the initial data $u$ and $v$. \par

We are following the same program in Carnot Groups. Our first goal is to study Hamilton-Jacobi equations in the subelliptic setting. For this purpose, we have to consider radial hamiltonians with a geometric distance, not equivalent to the Euclidean distance. We will establish in this new setting a version of the Hopf-Lax formula. This requires a careful examination of geodesics with respect to the new metric.\par Next,we present some definitions of convexity in the subriemannian setting. The notion of horizontal convexity has the right properties. In fact, upper-semicontinuous H-convex functions are Lipschitz and their symmetrized horizontal second derivatives are measures.

In addition, the corresponding H-semiconcavity is preserved under evolution via the simplest hamiltonian: the square of the distance.

Finally, we use the subelliptic Hopf-Lax formula to build semiconcave regularizations which preserve the subsolution or the supersolution character of the initial data.