Geometry And Analysis For A Class Of Nonlinear Elliptic Operater On Convex Domains

The convexity is an issue of interest for a long time in PDE, it is one of important geometric properties associated to the study of solutions of general elliptic partial differential equations, in particular for equations related to problems in differential geometry. The Constant Rank Theorem is a refined statement of convexity. This has profound implications in geometry of solutions of PDE and differential geometry. In this paper, we use Constant Rank Theorem to give some convexity results for a class of Hessian equation. At the same time, according to [21], we obtain the Brunn-Minkowski inequality in domain for the first eigenvalue of the corresponding Hessian operator, the point is the strict convexity result.The following are some convexity results obtained by constant rank theorem and deformation method. 0.4. Let Ω(?) R³ be a smooth bounded strictly convex domain. If u∈ C^∞(Ω|-) is a admissible solution of the following equation in Ωthen -(-u)^1/2 is strictly convex, and the convexity exponent 1/2 is sharp.定理 0.5. If u∈ C^∞(Ω) ∩ C^1,1(Ω|-) is an admissible solution of the following equation in a smooth bounded strictly convex domain Ω(?)R³then — log(—u) is strictly convex.Under the above regularity, we can also obtain the Brunn-Minkowski inequality for A, the first eigenvalue of the operator S₂, i.e. λ^-1/4 is concave in domain.0.6. Given any two convex bodies K₀, K₁ in R³ and t ∈ [0,1], λ satisfies the following inequalityλ((1 - t)K₀ + tK₁)^-1/4 ≥ (1 - t)λ (K₀)^-1/4 + tλ(K₁)^-1/4 (0.6)...