Extremal domains on Hadamard manifolds

Abstract

We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs.First, we study narrow properties of such domains in a Hadamard manifold and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity and how the domain behaves at infinity. This shows interesting relations with the Singular Yamabe Problem.Later, we focus on extremal domains in the Hyperbolic Space H n . Symmetry and boundedness properties will be shown. In certain sense, we extend Levitt-Rosenberg's Theorem [27] to OEPs, which suggests a strong relation with constant mean curvature hypersurfaces in H n . In particular, we are able to prove the Berestycki-Caffarelli-Nirenberg Conjecture under certain assumptions either on the boundary at infinity of the extremal domain or on the OEP itself.Also a height estimate for solutions on extremal domains in a Hyperbolic Space will be given. 0 MSC 2010: 35Nxx; 53Cxx.

Characterization of f-extremal disks

“…Regarding other Space Forms, combining the works of Molzon [25], Espinar-Mao [14] and Espinar-Farina-Mazet [12], we can prove the BCN conjecture for domains in the Hyperbolic space H 2 under similar hypothesis than the Euclidean case, specifically:…”

“…Theorem [12,14,25]. Let f be a non-increasing Lipschitz function and Ω ⊂ H 2 a C 2,α connected domain whose complement is connected.…”

Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

“…This method works also for the hyperbolic space and hemisphere [26] due to the existence of many totally geodesic hypersurfaces, but fails for more general geometries of nonconstant curvature. Other important symmetry or classification results concerning (mainly unbounded) G-extremal domains in spaces of constant curvature can be found in [6,11,12,13,32,39].…”

A Rigidity Result for Overdetermined Elliptic Problems in the Plane

“…In [36] the case of f -extremal epigraphs is solved for some nonlinearities f of the Allen-Cahn type. Finally, in [28] the result is proved if either f (t) ≥ λt or Ω is contained in a half-plane and ∇u is bounded (see also [13] for a generalization to other geometries). Observe that the assumption f (t) ≥ t excludes the prototypical Allen-Cahn nonlinearity; we point out that not even the half-plane is an f -extremal domain for those nonlinearities f .…”