The Convexity Estimates for the Solutions of Two Elliptic Equations

Abstract

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17] for the torsion equation and Acker et al. [1] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation v v = − 1 + v 2 in a smooth, bounded and strictly convex domain in n with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.

A note on parabolic power concavity

“…In this regard, convexity and convexity-like properties have been investigated extensively, especially in the field of elliptic equations. See [21] for a good introduction to this subject; see also [2,3,20,25,29] as some of the most recent results, to our knowledge, for elliptic equations. A typical question is the following: if u > 0 solves an elliptic Dirichlet problem in a bounded convex domain W with constant vanishing boundary data, does u have to be concave?…”

Concavity properties of solutions to Robin problems

“…Moreover, by Step 1, we have that w ≥ 0 also on ∂B ρ/2 provided we choose Proof. The result is due to C. Bandle [5], see also [31,Lemma 2.2].…”

“…-Estimate for η ∈ T ε (x 0 ). By (31), ( 32) and (37), we can choose a positive constant r 1 and a positive threshold…”

“…In quick terms, the reason is that such conditions allow establishing concavity near the boundary, which is a key step for any among the known methods to work. Yet, under Dirichlet boundary conditions, it is possible to go farther and obtain refined concavity estimates, such as the one given by Andrews-Clutterbuck in [2], with a far-reaching application to the proof of the fundamental gap conjecture; for different 'refinements' of concavity results, see also Ma-Shi-Ye [31] and Henrot-Nitsch-Salani-Trombetti [25]. Very recently, in the ground-breaking paper [3], Andrews-Cluttedbuck-Hauer have attacked the investigation of concavity properties under different boundary conditions, of Robin type.…”

Towards Optimal Gradient Bounds for the Torsion Function in the Plane

“…A fundamental result by Brascamp & Lieb [13] for the ground state of the Laplacian −∆u = λ 1 u inside convex domains in the plane is convexity of the level set. Problems of this nature are of continued interest [1,26,34,38,51,60,74]. We also emphasize that there are very precise results obtained for the structure of the first Laplacian eigenfunction by Grieser and Jerison [29,30,47] with exciting recent developments by T. Beck [8,9,10].…”