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By Robert Hardt, Michael Wolf

What distinguishes differential geometry within the final 1/2 the 20 th century from its prior historical past is using nonlinear partial differential equations within the research of curved manifolds, submanifolds, mapping difficulties, and serve as concept on manifolds, between different issues. The differential equations look as instruments and as gadgets of analysis, with analytic and geometric advances fueling one another within the present explosion of development during this quarter of geometry within the final two decades. This ebook comprises lecture notes of minicourses on the neighborhood Geometry Institute at Park urban, Utah, in July 1992. awarded listed below are surveys of breaking advancements in a couple of components of nonlinear partial differential equations in differential geometry. The authors of the articles should not basically very good expositors, yet also are leaders during this box of analysis. all the articles supply in-depth remedy of the themes and require few necessities and no more history than present examine articles.

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Extra resources for Nonlinear partial differential equations in differential geometry (Ias Park City Mathematics Series, Vol. 2)

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This step is carried over by means of approximation by solutions of the homogeneous problem (Lemma 11). A variation of Lemma 2 is the following: 24 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION Lemma 9. Let u be an element of S(f) in fl D B1. Assume that DI nQr, (x1) 36 0 and that Ilf II L"(B,) 5 8 (small enough), then there exist M and u, depending on A,,\, r1i r2,17, such that IDM nQr,(xi)I > 1z1Qr,(x1)I > 0. Proof. Let xo E D1 n Qr, (x1). Subtracting from u a linear function, we may suppose that the paraboloid at xo is 1-1x12.

Corollary 2. Assume that u is strictly convex, then u has a unique supporting plane at each point. Proof. If not we may assume that a) axn (a>0), b) u(0) = 0, c) u(-ten) = 0(t) (t > 0). We now consider the auxiliary function tl,,r =u-T(Xn+a). From the strict convexity of u, the set u,,r < 0 becomes compactly contained in the domain of definition of u for a, T positive and small. Also for T < a, u,,r attains its minimum at x = 0. Finally, we estimate the location of the two supporting planes to the set u,,r <0 of the form HI = {xn = C-},113 = {xn = C+}.

Indeed, consider a sequence uk for which sup h112(X - Xo) > 1 - 1/k. x#xo hl (X - Xo) From Lemma 1, Xo, {uk = 1/21 and {uk = 1} stay uniformly away from each other. In particular 0 < CI < C2. C1 IX -X01

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