Gaussian Curvature Estimates Of Level Sets Of Solutions To Some Elliptic Partial Differential Equations

As one of the most important geometric properties, convexity is an issue in the study of elliptic partial differential equations for a long time. In the present paper, we are concerned on the convexity of level sets of solutions to elliptic partial differential equations. By the classical maximum principle, we obtain the sharp positive lower bound estimates of the Gaussian curvature of the convex level sets of p-harmonic functions and minimal graphs in R". Under certain structural con-ditions, we also give the positive lower bound estimates of the Gaussian curvature of the convex level sets of solutions to some semi-linear elliptic equations. On the other hand, we establish the sharp concavity of the Gaussian curvature of the level sets of p-harmonic functions with respect to the height of the functions. More precisely, the main results of the paper are as follows.I. Gaussian curvature estimates of level sets of jy-harmonic functions Theorem0.0.1. Let Q be a bounded smooth domain in M.n(n≥2) and u∈C4(Ω)∩C2(Ω) be a p-harmonic function, i.e. div(|â–½u|p-2â–½u)=0in Ω. Assume1<p<+oo,|â–½u|≠0on Ω. Let K be the Gaussian curvature of the level sets. If the level sets of u are strictly convex with respect to the direction of gradientâ–½u, then we have the following statements.Case1; For n≥2,1<p<+∞; the function|â–½u|n+1-2pK attains its minimum on the boundary.Case2: For n=2,1<p<+∞; and for n≥3,1+2/n≤p≤n, the function|â–½u|1-pK attains its minimum on the boundary.Case3:For n=2,2/3≤p≤3; n=3,2≤p<+∞or n≥4, p=n+1/2, the function K attains its minimum on the boundary.By Theorem0.0.1, we can obtain the Gaussian curvature estimates of level sets of p-harmonic functions. Corollary0.0.2. Let u satisfy the following Dirichlet problem where1<p<+∞; Ω0and Ω1are bounded smooth convex domains in Rn(n≥2), Ω1(?)Ω0.Let K be the Gaussian curvature of the level sets. Then we have the following estimates.Case1a: For1<p≤n+1/2, we haveCase1b: Forn+1/2<p<+∞; we haveCase2: For n=2,1<p<+∞; and for n≥3,1+2/n≤p≤n, we haveCase3:Forn=2,3/2≤p≤3; n=3,2≤p<+∞or n≥4, p=n+1/2,we have Ωmin K≥δΩmin K. In particular, for the harmonic functions we have the following proposition. Proposition0.0.3. Let Ω Q be a domain in Rn(n≥2).Let u be a harmonic function defined in Q. Assume|â–½u|≠0in Ω. Let K be the Gaussian curvature of the level sets. Let ψ=|â–½u|-K.If the level sets of u are strictly convex with respect to the direction of gradientâ–½u, then the function ip is a super-harmonic function modulo the gradient terms▽ψ. Namely, we have the following differential inequality ε▽ψ≤C|▽ψ|in Ω, where the positive constant C depends on n and||u||c3(Ω)â…¡. Gaussian curvature estimates of level sets of minimal surfaces Theorem0.0.4. Let Ω1be a smooth bounded domain in Rn(n≥2). Let u∈C4(Ω)(?)C2(Ω1) satisfy the following minimal surface equation Assume|â–½u|≠0on Ω. Let K be the Gaussian curvature of the level sets. If the level sets of u are strictly convex with respect to the direction of gradient Vu, then we have the following results.(â…°) For n=2, the function K attains its minimum and max-imum on the boundary.(â…±) For n≥3, the function attains its minimum on the boundary forIn a similar way, we have the Gaussian curvature estimates of level sets of minimal surfaces. Corollary0.0.5. Let u satisfy the following Dirichlet problemwhere Ω0Ω1are bounded smooth convex domains in Rn(n≥2),Ω1(?)Ω0. Let K be the Gaussian curvature of the level sets. Then we have the following estimatesâ…¢. Gaussian curvature estimates of level sets of solutions to semi-linear elliptic equations Theorem0.0.6. Let Ωbe a bounded smooth domain in Rn(n≥2). Let u∈C4(Ω)∩C2(Ω1) satisfy the following semi-linear elliptic equationâ–½u=f(x,u,â–½u) in Ω,where f≥0,f∈C2(Ω×R×Rn). Assume|â–½u|≠0on Ω. Let K be the Gaussian curvature of the level sets. Assume the level sets of u are strictly convex with respect to the direction of gradientâ–½u. Denote the following two assertions by (A1) and (A2), i.e.(A1) The function|â–½u|-2K attains its minimum on the boundary;(A2) The function|â–½u|n-1K attains its minimum on the boundary. Then we have the following facts.Case1: f=f(u). If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid.Case2: f=f(x). If the map F:(0,+∞)×Ωâ†'R,(t, x) hâ†'t3f(x) is convex or f-1/2is concave for positive f), then (A2) is valid. Case3:f=f(x,u). Assume for every choice of u∈(0,1), the map Fu:(0,+∞)×Ω×R,(t, x)â†'t3(x, u)is convex. If fu≤0, then (A2) is valid. Case4:f=f(it,â–½u). Assume/or e^/ery choice of u∈(0,1), the map Fu:(0,+∞)×Sn-1â†'R,(t, p)â†'t3f(u,p/t)is convex. If fu≥0, then (A1) is valid; if fu≤0, then (A2) is valid. Case5:f=f(x,u,â–½u).Assume for every choice of u∈(0,1), t/je map Fu:(0,+∞)×Ω×Sn-1â†'R,(t, x,p)â†'t3f(x,u,p/t)is convex. If fu≤0, i/ien (A2) zCorollary0.0.7. Lei u satisfy the following Dirichkt problemwhere Ω0and Ω1are bounded smooth convex domains in Rn(n≥2), Ω1×Ω0andf∈C2(R), nondecreasing,f(0)=0. Let K be the Gaussian curvature of the levelsets. Then we have the following estimates â…£. Concavity of the Gaussian curvature of level sets ofp-harmonicfunctions with respect to the height of the functionsTheorem0.0.8. Let u satisfy the following Dmchlet problem Xllwhere Ω0and Ω1are bounded smooth strictly convex domains in Rn(n≥2),Ω1(?) Ω0and1<p<+∞.Fort∈(0,1), denote Ωt={x∈Ω: u(x)=t}. Let K be the Gaussian curvature of the level sets. For t∈[0,1], define the function f(t)=x∈Ωt/min(|â–½u|n+1-2pK)1/n-a(x).Then f(t) is a concave function for t∈[0,1], Equivalently, for any point x∈Ωt, we have the following estimates (|â–½u|n+1-2pK)1/n-1(x)≥(1-t)δΩ0max(|â–½u|n+1-2pK)1/n-1=tδΩ1max(|â–½u|n+1-2pK)1/n-1.Furthermore, the function f(t) is an affine function of the height t when the p-harmonic function is the p-Green function on the ball.