Uniform Level Set Estimates for Ground State Eigenfunctions

We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach the maximum, uniformly with respect to the eccentricity of the domain itself. This is achieved by obtaining quantitatively sharp second derivative estimates, which are consistent with the shape of the superlevel sets. In particular, we prove that the eigenfunction is concave (rather than merely log concave) in an entire superlevel set near its maximum. By estimating the mixed second and third derivatives partial derivatives of the eigenfunction, we also determine the rate at which a degree 2 Taylor polynomial approximates the eigenfunction itself.

[1]  P. Freitas,et al.  Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the laplacian on thin planar domains , 2007, 0712.3245.

[2]  Leonid Friedlander,et al.  On the Spectrum of the Dirichlet Laplacian in a Narrow Infinite Strip , 2008 .

[3]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[4]  Nicholas J. Korevaar.  Convexity Properties of Solutions to Elliptic P.D.E.’S , 1987 .

[5]  Leonid Friedlander,et al.  On the spectrum of the Dirichlet Laplacian in a narrow strip , 2007 .

[6]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[7]  Location of the nodal set for thin curved tubes , 2006, math/0602470.

[8]  The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schr\"odinger Operators , 2014, 1411.7353.

[9]  Ben Andrews,et al.  Proof of the fundamental gap conjecture , 2010, 1006.1686.

[10]  David Jerison,et al.  The diameter of the first nodal line of a convex domains , 1995 .

[12]  John L. Lewis,et al.  Convex solutions of certain elliptic equations have constant rank hessians , 1987 .

[13]  E. Lieb,et al.  On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , 1976 .

[14]  F. John.  Extremum Problems with Inequalities as Subsidiary Conditions , 2014 .

[15]  Level Set Shape For Ground State Eigenfunctions On Convex Domains , 2015 .

[16]  Daniel Grieser,et al.  The size of the first eigenfunction of a convex planar domain , 1998 .

[17]  M. Solomyak,et al.  On the spectrum of narrow periodic waveguides , 2008 .

[18]  A. Friedman,et al.  Convexity of solutions of semilinear elliptic equations , 1985 .

[19]  Daniel Grieser,et al.  Asymptotics of the first nodal line of a convex domain , 1996 .