Calculus of Variations and Partial Differential Equations: by Luigi Ambrosio

By Luigi Ambrosio

The hyperlink among Calculus of diversifications and Partial Differential Equations has continuously been powerful, simply because variational difficulties produce, through their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can frequently be studied by means of variational equipment. on the summer season college in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each one gave a direction on a classical subject (the geometric challenge of evolution of a floor by means of suggest curvature, and measure concept with functions to pde's resp.), in a self-contained presentation obtainable to PhD scholars, bridging the distance among regular classes and complex learn on those issues. The ensuing publication is split hence into 2 elements, and well illustrates the 2-way interplay of difficulties and strategies. all the classes is augmented and complemented through extra brief chapters by means of different authors describing present learn difficulties and results.

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Proof. (i) As in Theorem 15, it is not restrictive to assume that fl and u are convex. Since we need only to prove that D{{u is a Radon measure for any unit vector ~ ERn. A smoothing argument shows that Deeu ~ 0 in the sense of distributions. Let Ace fl and let q, E C~ (fl) be a function equal to 1 in a neighbourhood of Aj for any cp E C~(A) it holds This shows that the distribution is bounded in A, hence locally bounded in fl. (ii) In the proof of this statement we make a stronger (and not restrictive) assumption, namely D 2 u ~ I.

49) We will prove now that this equation actually characterizes the distance function: Theorem 12 (viscosity characterization of distance functions). Let C c Rn be a closed set and let A := Rn \ C. Let u E C(A) be a nonnegative viscosity solution of the equation (49) vanishing on 8A. Then, C is not empty and u(x) = dist (x, C) VxEA. Proof. Let w(x) = dist (x, C) and extend u to all Rn setting u = 0 on C; we have to prove that u == w. We will first assume that A is bounded. It is easy to check that l'Vul 2 - 1 ~ 0 in the whole Rn, in the viscosity sense, so that Lemma 1 below yields that the Lipschitz constant of u is less than 1.

Remark 13. Obviously, any C 2 perturbation of a convex function is semiconvex, and the semiconvexity constant can be estimated with the C 2 norm of the perturbation. The definitions of semiconcave function and of semiconcavity constant can be given in a similar way. An important example of semiconcave function is the squared distance function from any nonempty set E: indeed, the identities show that dist 2 (x, E) is semiconcave in Rn with semiconcavity constant less than 2. Exercise 6. e. { in u {j2¢ ( aeae dx ~ -c in ¢dx Moreover, the smallest constant c (59) ~ 0 in (59) is sc(u, n).

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