Dirichlet duality and the nonlinear Dirichlet problem
By F. R. Harvey, H. Lawson. 2007
215 citations recorded by the source corpus.
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ⋐ ℝn. In our approach the equation is replaced by a subset F ⊂ Sym2(ℝn) of the symmetric n × n matrices with ∂F ⊆ {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ∂Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F̃ that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F̃ is F, and in the analysis the roles of F and F̃ are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ℝ, ℂ, and ℍ; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c0c060fef25df50e8b6a15ce2bb68142a29e018e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- An introduction to potential theory in calibrated geometry (2007, background): “scarce, the partially pluriharmonic functions represent the calibrated analogue of solutions to the homogeneous Monge-Amp´ere equation, and they are sufficiently abundant to solve the Dirichlet Problem [HL3,6]. For now we mention a “fundamental” example. P OPOSITION R 1.7. Suppose φ∈ ΛpRnis a parallel”
- Singular Solution to Special Lagrangian Equations (2009, methodology): “Recently a new very interesting approach to degenerate elliptic equations was suggested by Harvey and Lawson [HL2].”
- Games for eigenvalues of the Hessian and concave/convex envelopes (2018, methodology): “…a strict minimum at the point x ∈ Ω with u ∗ ( x ) = φ ( x ) , we have (2) for every ψ ∈ C 2 such that u ∗ − ψ has a strict maximum at the point x ∈ Ω with u ∗ ( x ) = ψ ( x ) , we have Now, we refer to [6] for the following existence and uniqueness result for viscosity solutions to ( λ j , g ).”
- A family of degenerate elliptic operators: Maximum principle and its consequences (2016, unknown): “More recently, in a PDE context, we wish to recall the works of Harvey and Lawson [20,21] that have given a new geometric interpretation of solutions, while Caffarelli, Li and Nirenber g in [12,13] in their study of degenerate elliptic equations, give some results concerning removable singularities…”
- Singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems (2011, background): “roach toward the existence and uniqueness of C0 viscosity or weak solution for the Dirichlet problem to strictly elliptic as well as degenerate elliptic fully nonlinear equations by Harvey and Lawson [HL2] [HL3]. Recently Nadirashvili and Vl˘aduc¸t [NV] constructed beautiful C1,1/3 singular viscosit”