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Consider a locally Lipschitz function $u$ on the closure of a possibly unbounded open subset $\Omega$ of $\mathbb{R}^n$ with $C^{1,1}$ boundary. Suppose $u$ is semiconcave on $\overline \Omega$ with a fractional semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points., Comment: 15 pages, no figures

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point., 18 pages

For $\alpha\in (0,2)$ we study the null controllability of the parabolic operator $Pu= u_t - (\vert x \vert ^\alpha u_x)_x \ (-1

We consider the transport equation \(\partial _t u(x,t) + H(t)\cdot \nabla u(x,t) = 0\) in \(\varOmega \times (0,T),\) where \(T>0\) and \(\varOmega \subset \mathbb R^d \) is a bounded domain with smooth boundary \(\partial \varOmega \). First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of H intersect \(\partial \varOmega \), we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.

The regularity of the gradient of viscosity solutions of first-order Hamilton-Jacobi equations partial derivative(t)u(t,x) + H(t, x, D(x)u(t,x)) = 0; t is an element of R+, x is an element of R-n, u(0,x)= u(0)(x), x is an element of R-n, is studied under a strict convexity assumption on H(t,x,). Estimates on the discontinuity set of Du are derived. Such estimates imply that solutions of the above problem are smooth in the complement of a closed H-n-rectifiable set. In particular, it follows that Du belongs to the class SBV, i.e., D(2)u is a measure with no Canter part.

Given a semi-convex functionu: ω⊂R n→R and an integerk≡[0,1,n], we show that the set ∑k defined by $$\Sigma ^k = \left\{ {x \in \Omega :dim\left( {\partial u\left( x \right)} \right) \geqslant k} \right\}$$ is countably ℋn-k i.e., it is contained (up to a ℋn-k-negligible set) in a countable union ofC 1 hypersurfaces of dimensions (n−k). Moreover, we show that $$\int\limits_{\Omega ' \cap \Sigma ^k } {\mathcal{H}^k } \left( {\partial u\left( x \right)} \right)d\mathcal{H}^{n - k} \left( x \right)< + \infty $$ for any open set ω′⊂⊂ω.