Contemporary Mathematics Volume 570, 2012 Geometric PDEs in the presence of isolated singularities Jos´ e A. alvez and Pablo Mira Abstract. This is a short course on the behavior of solutions to some geo- metric elliptic PDEs of Monge-Amp` ere type in two variables, in the presence of non-removable isolated singularities. We will describe local classification theorems around such an isolated singularity, as well as global classification theorems for the case of finitely many isolated singularities. 1. Introduction Given a smooth solution u = u(x, y) of an elliptic partial differential equation (PDE) on a punctured disc D∗ = {(x, y) R2 : 0 (x−a)2 +(y −b)2 r2}, a clas- sical problem is to describe the behavior of u at the puncture q = (a, b). The usual question in this setting is whether the singularity is removable, i.e. whether the solution u C2(D∗) actually extends smoothly to D. For some classical geometric equations like the minimal graph equation (1 + ux)uyy 2 2uxuyuxy + (1 + uy)uxx 2 = 0, any isolated singularity is automatically removable (see [Ber]). This results holds in a much wider context, and it has been generalized by many authors. For instance, very recently by Leandro and Rosenberg [LeRo] for the prescribed mean curvature equation associated to a Killing submersion. For many other quasilinear elliptic equations, isolated singularities are removable as long as u lies in the Sobolev space H1(D∗) W 1,2 (D∗), see for instance [GiTr]. Contrastingly, some elliptic PDEs admit solutions u(x, y) with non-removable isolated singularities, such that both the solution and its first derivatives are bounded around the singularity but u does not extend C2 across it. For example, let us consider the general elliptic Monge-Amp` ere equation in dimension two: (1.1) uxxuyy uxy 2 = F (x, y, u, ux,uy) c 0, where F is a smooth function. Observe that if u(x, y) is a solution to this equation, then its graph z = u(x, y) is a locally convex surface in R3. In these conditions, any solution to (1.1) on a punctured disk extends continuously to the puncture, but this extension is not necessarily C1-smooth at this point: it could have a conical singularity at the puncture (see Figure 1). 1991 Mathematics Subject Classification. Primary 53C42, 53C45, 35J96 Secondary 35J60. Key words and phrases. Isolated singularities, conical singularities, Monge-Amp` ere equation, surfaces of constant curvature, improper affine spheres, harmonic maps. c 2012 American Mathematical Society 1
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