By Matthew J. Gursky, Ermanno Lanconelli, Andrea Malchiodi, Gabriella Tarantello, Xu-Jia Wang, Paul C. Yang, Antonio Ambrosetti, Sun-Yung Alice Chang

This quantity includes the notes of the lectures added on the CIME path GeometricAnalysis andPDEsduringtheweekofJune11–162007inCetraro (Cosenza). the varsity consisted in six classes held by way of M. Gursky (PDEs in Conformal Geometry), E. Lanconelli (Heat kernels in sub-Riemannian s- tings),A. Malchiodi(Concentration of options for a few singularly perturbed Neumann problems), G. Tarantello (On a few elliptic difficulties within the examine of selfdual Chern-Simons vortices), X. J. Wang (Thek-Hessian Equation)and P. Yang (Minimal Surfaces in CR Geometry). Geometric PDEs are a ?eld of analysis that is at the moment very lively, because it makes it attainable to regard classical difficulties in geometry and has had a dramatic impression at the comprehension of 3- and 4-dimensional ma- folds within the final a number of years. On one hand the geometric constitution of those PDEs may well reason normal di?culties as a result presence of a few invariance (translations, dilations, selection of gauge, and so forth. ), which leads to an absence of c- pactness of the practical embeddings for the areas of services linked to the issues. nevertheless, a geometrical instinct or end result may well give a contribution vastly to the hunt for normal amounts to maintain tune of, andtoproveregularityoraprioriestimatesonsolutions. Thistwo-foldaspect of the examine makes it either tough and intricate, and calls for using severalre?nedtechniquestoovercomethemajordi?cultiesencountered.

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**Additional resources for Geometric Analysis and PDEs: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 11-16, 2007**

**Sample text**

E. e. e. for every x, y ∈ RN there exists γ : [0, 1] → RN , continuous and such that d(x, y) = d(x, γ(t)) + d(γ(t), y) for every t ∈ [0, 1] We shall denote by |H| the constant |H| := Λ + a0 + b0 + cd Under the previous assumptions, we show the following scale invariant Harnack inequality which extends to our sub-Riemannian setting the classical Hadamard-Pini parabolic Harnack inequality Harnack inequality. If Hu = 0 and u ≥ 0 in an open set containing [t0 − R2 , t0 ] × B(x0 , R), where (t0 , x0 ) =: z0 ∈ S, then max u ≤ M u(z0 ).

N (p) are real and independent of B and the deﬁning function f of D. So that they only depend on the domain D. Heat Kernels in Sub-Riemannian Settings 39 Just proceeding as in the real case, one can deﬁne the m−th Levi curvature of ∂D at p, 1 ≤ m ≤ n, as Kpm (∂D) = σ (m) (λ1 (p), . . , λn (p)) n m , where σ (m) denotes the m-th elementary symmetric function. , λn (p)) . g. in [11]. In linearized form, the equations of this new class can be written as (see [29, equation (34) p. 324]) 2n Lu ≡ aij Du, D2 u Xi Xj u = K (x, u, Du) inR2n+1 (1) i,j=1 where: the Xj ’s are ﬁrst order diﬀerential operators, with coeﬃcients depending on the gradient of u, which form a real basis for the complex tangent space to the graph of u; the matrix {aij } depends on the function s; K is a prescribed function.

Yang Paul C. ; Chang. ; rsted, Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), 669–682. [Bra96] Thomas P. Branson, An anomaly associated with 4-dimensional quantum gravity, Comm. Math. Phys. 178 (1996), 301–309. [CGY99] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, Regularity of a fourth order nonlinear PDE with critical exponent, Amer. J. Math. 121 (1999), no. 2, 215–257. [CGY02a] Sun-Yung A. Chang, Matthew J. Gursky, and Paul Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J.