By Matthew J. Gursky, Ermanno Lanconelli, Andrea Malchiodi, Gabriella Tarantello, Xu-Jia Wang, Paul C. Yang, Antonio Ambrosetti, Sun-Yung Alice Chang
This quantity comprises the notes of the lectures introduced on the CIME path GeometricAnalysis andPDEsduringtheweekofJune11–162007inCetraro (Cosenza). the college consisted in six classes held by means of M. Gursky (PDEs in Conformal Geometry), E. Lanconelli (Heat kernels in sub-Riemannian s- tings),A. Malchiodi(Concentration of suggestions for a few singularly perturbed Neumann problems), G. Tarantello (On a few elliptic difficulties within the learn 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 study that's presently very energetic, because it makes it attainable to regard classical difficulties in geometry and has had a dramatic effect 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 reason basic di?culties end result of the presence of a few invariance (translations, dilations, collection of gauge, and so on. ), which ends up in an absence of c- pactness of the useful embeddings for the areas of features linked to the issues. however, a geometrical instinct or consequence may perhaps give a contribution drastically to the quest for normal amounts to maintain tune of, andtoproveregularityoraprioriestimatesonsolutions. Thistwo-foldaspect of the examine makes it either demanding and intricate, and calls for using severalre?nedtechniquestoovercomethemajordi?cultiesencountered.