Convexity Estimates For Solutions Some Elliptic Partial Differential Equations And Their Applications

The geometric property of solutions is a fundamental problem in the theory of partial differential equations. As one of the most important geometric properties, con-vexity has been an issue in the study of elliptic partial differential equations for a long time. In the present paper, we are concerned on the convexity estimates for solutions to elliptic partial differential equations. By the classical maximum principle, we ob-tain the convexity estimates for the solution to the Saint-Venant torsion problem, for the first eigenfunction and the Green’s function of the Laplace operator in a convex bounded domain. On the other hand, we also research the Gauss curvature and mean curvature estimates for the level sets of solutions to Monge-Ampere equation. In detail, the main results of the paper are as follows.Ⅰ. Convexity estimates for the solution of the Saint-Venant torsion problem Theorem0.1. Let Ω be a smooth, bounded convex domain in Rn, n≥2, and u the solution for the Saint-Venant torsion problem Ifv=-(?) is a strictly convex function, then the function satisfies the following elliptic differential inequality: Δψ1≤0mod ((?)ψ1) in Ω,where we have suppressed the terms containing the gradient of ψ1with locally bounded coefficients. Moreover, the function ψ1attains its minimum on the boundary (?)Ωand has the following estimatewhere K is the Gaussian curvature of (?)Ω. Finally, the function ψ1attains its minimum in Ω if and only ifΩ is an ellipsoid(ellipse).Corollary0.2. Let Ω be a smooth, bounded and strictly convex domain in Rn, kmin, kmax and Kmin the minimal principal curvature, maximal principal curvature and the minimal Gaussian curvature of the boundary (?)Ω respectively. If u is the solution for the Saint-Venant torsion problem and v=-(?), then Gaussian curvature KG for the graph of v satisfies the following sharp estimateWhen we take Ω being the unit ball B1(0)(?)Rn, the equality above holds at the origin0.II. Convexity estimates for the first eigenfunction of Laplace operator Theorem0.3. Let Ω be a smooth, bounded convex domain in Rn, n≥2, and u>0the solution for the first eigenvalue problem of Laplace operator; If v=-log u is a strictly convex function, then the function satisfies the following elliptic differential inequality: Δψ2≤0mod ((?)ψ2) in Ω,where we have suppressed the terms containing the gradient of ψ2with locally bounded coefficients. Moreover, the function ψ2attains its minimum on the boundary and has the following estimate where K is the Gaussian curvature of (?)Ω.Ⅲ. Convexity estimates for the Green’s functionTheorem0.4. Let Ω be a smooth, bounded convex domain in Rn, n≥2, x0∈Ω. Let u>0be the Green’s function of Ω with pole at the point x0,i.e. the solution for the problemwhere δ(x-x0) is the Dirac measure at the point x0. Let v=e-αu,α>2π for n=2or v=u1/2-n for n≥3. If v is a strictly convex function, then the function ψ=v2-n2det D2v that is, or satisfies the following elliptic differential inequality: Δψ≤0mod ((?)ψ) in Ω\{x0},where we have suppressed the terms containing the gradient of ψ with locally bounded coefficients. Moreover, the function ψ attains its minimum on the boundary (?)Ω and has the following estimate for n≥3: where K is the Gaussian curvature of (?)Ω.Corollary0.5. Let Ω be a smooth, bounded and strictly convex domain in Rn, n≥3. x0∈Ω. Let u>0be the Green’s function of Ω with pole at the point x0and v=u1/2-n. Then v is strictly convex in Ω\{x0}. IV. Gauss curvature and mean curvature estimates for the level sets of solu-tions to Monge-Ampere equationTheorem0.6. Let Ω be a bounded convex domain in Rn, n≥2. If is the solution for the following Dirichlet problem of Monge-Ampere equation: then the function attains its maximum on the boundary (?)Ω.Theorem0.7. Under the same assumptions in the above Theorem, the functionattains its maximum on the boundary (?)Ω. Moreover, ψ attains its maximum in Ω if and only if Ω is an ellipse for n=2or a ball for n≥3.Corollary0.8. Let Ω be a smooth, bounded and strictly convex domain in Rn, n≥2. If u is the solution for the Dirichlet problem of Monge-Ampere equation in the above Theorem, then the functions K|(?)u|n+1and H|(?)u|3attain their maximum only at the boundary (?)Ω. Thus, for x∈Ω\Ω’, we have the following estimates andwhere Ω’={x∈Ω|u(x)<c,c∈(minΩ u,0) is a constant}, K,H,Km,kM are Gauss curvature, mean curvature, minimal and maximal principal curvature of the level sets at a point respectively.