By Luigi Ambrosio

The hyperlink among Calculus of adaptations and Partial Differential Equations has regularly been robust, simply because variational difficulties produce, through their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can frequently be studied through variational tools. on the summer season college 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 through suggest curvature, and measure idea with purposes to pde's resp.), in a self-contained presentation available to PhD scholars, bridging the distance among ordinary classes and complex study on those themes. The ensuing e-book is split for that reason into 2 elements, and well illustrates the 2-way interplay of difficulties and techniques. all of the classes is augmented and complemented by means of extra brief chapters by way of different authors describing present examine difficulties and results.

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**Additional resources for Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory**

**Example text**

Then, problem (83) has a continuous viscosity solution u satisfying f ::; u ::; g. Proof. By Theorem 11, there exists a viscosity solution w of (71) in (0, T) x Rn lying between f and g. Since w· ::; g* = '110 = f* ::; w* at time 0, by applying Theorem 18 with u := w*, v := w. , w is continuous. 0 Remark 18. By applying Theorem 18 to any pair of solutions, we obtain that the solution of (83) is also unique in the class of functions with linear growth. Hence, existence of solutions of (83) is reduced to a construction of suitable subsolutions and supersolutions of the problem which satisfy the initial condition.

Proof. Assume first that u is bounded and () is smooth with ()' > 0 on R. Let 1/J be the inverse function of ()j if (){u) - ' = 1/J'{

Let p E Rn \ {O} and let Pp = I - P ® p/lpl2 be the orthogonal projection on the hyperplane pl. orthogonal to p. ). Then, we define ° k (84) Gk(P,X):= - LA;(Y) ;=1 where Al (Y), ... , in increasing order (in other words, we remove the top (n - 1 - k) eigenvalues). In the case k = (n-l), which corresponds to the mean curvature flow for hypersurfaces, G k (p, X) reduces to -trace Y = -trace (PpX Pp) = -trace (PpX) = -trace X + (Xp,p) Ip12' (85) By (15) and Remark 4, the equation corresponding to this choice of G k flows each co dimension 1 level set with velocity equal to the sum of the smallest k principal curvatures.