By Luigi Ambrosio

The hyperlink among Calculus of diversifications and Partial Differential Equations has consistently been powerful, simply because variational difficulties produce, through their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can frequently be studied via variational tools. on the summer time university in Pisa in September 1996, Luigi Ambrosio and Norman Dancer every one gave a direction on a classical subject (the geometric challenge of evolution of a floor by way of suggest curvature, and measure thought with purposes to pde's resp.), in a self-contained presentation available to PhD scholars, bridging the space among ordinary classes and complex learn on those issues. The ensuing e-book is split for this reason into 2 elements, and well illustrates the 2-way interplay of difficulties and strategies. all of the classes is augmented and complemented via extra brief chapters by means of different authors describing present study difficulties and results.

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**Example text**

Since Xh E B for h large enough, it holds liminf fh(Xh) 2: lim infinf fh h-+oo h-+oo B = c(B). Since B is arbitrary, (53) follows. To prove (54) we choose a sequence of balls Bk 3 x belonging to U and such that C(Bk) t f(x). h = C(Bk) h-+oo ~ "Ik E N. h h-+oo ~ 00 such that f(x). Bk(h) Choosing Xh E Bk(h) such that fh(Xh) ~ inf fh Bk(h) + 2- h, (54) follows. 0 36 Part I, Geometric Evolution Problems Exercise 5. Let X be a locally compact metric space and let fh, I : X --t R be functions. Show that (fh) converges to I locally uniformly if and only if r- h-+oo lim Ih = I = r+ lim fh· h-+oo Theorem 13 (variational properties of r- limits).

And F* satisfy (89) for any pair (P,X). X) = )"F(p, X) 'V).. F(p,X +ap®p) Noticing that 'Va E R we obtain that any F(p, X) which actually depends only on Y, as the function Gk in (84), fulfils the second condition in (89). It is evident that Gk satisfies the first one too. Another example of geometric evolution equation is Ut - t QI Vul trace (Pvu IVul V 2 u Pvu ){3 = O. In the two dimensional case, this equation has been studied by Alvarez, Lions and Morel (see [ALCM93, ALM92]) as a model for non isotropic smoothing of images.

Q it holds Proof. Inequality ~ simply follows choosing as L the subspace spanned by the eigenvectors corresponding to Ai(Y)"'" Aq(Y), because Y 2': Ai(Y)J on L. To prove the opposite inequality, let L be any subspace with dimension at least q - i + 1 and let M be the i-dimensional vector space spanned by the eigenvectors corresponding to Al (Y), ... , Ai (Y). Then, choosing a unit vector vEL n M we find because Y ~ o Ai(Y)J on M. Now we will construct subsolutions and supersolutions for u(O,·) = Uo.