Proposing the lawsuits of the twenty-first Nordic Congress of Mathematicians at Luleå college of expertise, Sweden, this striking reference discusses fresh advances in research, algebra, stochastic methods, and using pcs in mathematical study.

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Example text

Thus, the distance from C is the inﬁmum of the distance from such spheres. 5 it sufﬁces to prove the result in the case C = B r . We have, for any x ∈ Rn \ C, dC (x) = |x| − r = d{0} (x) − r. Since the distance between {0} and the complement of C is r , we obtain from part (ii) that dC is semiconcave with the desired modulus. (iv) Let y be any point in the complement of C and let x be a projection of y onto C. Then, if we set y−x ν= |y − x| we have that dC (x + hν) = h for all h ∈ [0, dC (y)]. Therefore, since dC is nonnegative, we obtain dC (x + hν) + dC (x − hν) − 2dC (x) ≥ h, showing that dC is not semiconcave in any neighborhood of x.

The chapter is structured as follows. 1 we deﬁne semiconcave functions in full generality, and study some direct consequences of the deﬁnition, like the Lipschitz continuity and the relationship with the differentiability. 2, like the distance function from a set, or the solutions to certain partial differential equations. We give an account of the vanishing viscosity method for Hamilton–Jacobi equations, where semiconcavity estimates play an important role. 3 we recall some properties which are peculiar to semiconcave functions with a linear modulus, like Alexandroff’s theorem or Jensen’s lemma.

A) We have D + u(x) = { p ∈ Rn : ∂ + u(x, θ ) ≤ p, θ ∀ θ ∈ Rn }, D − u(x) = { p ∈ Rn : ∂ − u(x, θ ) ≥ p, θ ∀ θ ∈ Rn }. (b) D + u(x) and D − u(x) are closed convex sets (possibly empty). (c) D + u(x) and D − u(x) are both nonempty if and only if u is differentiable at x; in this case we have that D + u(x) = D − u(x) = {Du(x)}. 1). The converse can be proved by contradiction. Indeed, / D + u(x). 7) for all k ∈ N. Moreover, possibly taking a subsequence, we may assume that the sequence of unit vectors xk − x θk := |xk − x| converges to some unit vector θ as k → ∞.