By Enrico Giusti

This e-book presents a complete dialogue at the life and regularity of minima of normal integrals within the calculus of adaptations and of recommendations to elliptic partial differential equations and platforms of the second one order. whereas direct tools for the life of ideas are renowned and feature been normal within the final century, the regularity of the minima used to be regularly received through the Euler equation as part of the overall thought of partial differential equations. during this ebook, utilizing the idea of the quasi-minimum brought by means of Giaquinta and the writer, the direct equipment are prolonged to the regularity of the minima of functionals within the calculus of adaptations, and of recommendations to partial differential equations. This unified remedy deals a considerable financial system within the assumptions, and allows a deeper knowing of the character of the regularity and singularities of the recommendations. The e-book is basically self-contained, and calls for just a basic wisdom of the weather of Lebesgue integration thought.

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If R < Ro we have therefore for every £ e R w : / \u-Z\*dy= Jn(y0,R) \U - i\p\J\dx < c f [ •/fl- 1 ("(»o,fl)) \U-Z\vdx JA(xo,LR) from which the inequality HHI| P ,A

27) LAPLACE n A=:^AA. Proof. Let xo £ £ r and let yo be the point corresponding to XQ on <9fi. We can assume that yo = 0 and that the tangent plane to <9fi in 0 is the horizontal plane xn = 0. (0) = 0; we can assume that Q, lies locally below the graph of 1?. , -«n-i] • (1-28) Semi-Classical 27 Theory Let V be the intersection of W with the plane xn = 0; in V x R we define •y(y,d) = y + u(y)d; y = (y,d(y)). We have 7 £ C f c _ 1 ; recalling that Vi{y) =-VnD^y) (i = 1,2, . . , 1 - K n _id, 1]. 29) If d < T, the Jacobian determinant Dj is positive, and therefore it is possible to express y and d as functions of x, both of class C f e _ 1 in a neighborhood of x^ =: (0,d).

6 Let Q, be a bounded open set in R n , whose boundary is a C2 manifold with non-negative mean curvature. 41) has a unique minimum in Lip (fi, U). In particular, the DlRlCHLET problem has a solution for every C2 boundary datum if Q, is convex. However, except in the two-dimensional case (n = 2), the condition of non-negative mean curvature is obviously more general than that of convexity. 46) is satisfied only if we assume that Ad < eo, with eo > 0 depending only on the C2 norm of the function U.