[Paper Review] Towards a reversed Faber-Krahn inequality for the truncated Laplacian

Isabeau Birindelli, Giulio Galise+1 more|arXiv (Cornell University)|Mar 20, 2018

Nonlinear Partial Differential Equations18 references6 citations

TL;DR

This paper investigates the principal eigenvalue problem for the truncated Laplacian operator $\mathcal{P}^{+}_{k}$, a degenerate fully nonlinear elliptic operator. It establishes a reversed Faber-Krahn inequality: among rectangles of fixed volume, the square maximizes the principal eigenvalue $\mu_1^+$, contradicting the classical Faber-Krahn inequality for the Laplacian, where the ball minimizes the eigenvalue. The result is proven via explicit construction of eigenfunctions using one-dimensional ODEs and convex analysis.

ABSTRACT

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain maximize the principal eigenvalue, contrary to what happens for the Laplacian.

Motivation & Objective

• To investigate whether symmetry of the domain maximizes the principal eigenvalue for the truncated Laplacian $\mathcal{P}^{+}_{k}$, contrary to the classical Faber-Krahn inequality for the Laplacian.
• To determine if strict convexity is necessary for the existence of a positive eigenfunction for $\mathcal{P}^{+}_{1}$, especially in non-strictly convex domains like rectangles.
• To explore whether qualitative properties of the Laplacian's principal eigenvalue, such as monotonicity and symmetry effects, extend to the truncated Laplacian.
• To establish a reversed Faber-Krahn inequality for $\mathcal{P}^{+}_{1}$, showing that among rectangles of fixed volume, the square yields the largest eigenvalue.

Proposed method

• The authors define the truncated Laplacian $\mathcal{P}^{+}_{k}(D^2u) = \sum_{i=1}^k \lambda_{N+1-i}(D^2u)$, where $\lambda_i$ are the ordered eigenvalues of the Hessian matrix.
• They construct explicit eigenfunctions for $\mathcal{P}^{+}_{1}$ in rectangles as products of one-dimensional functions, using solutions to ODEs involving the largest eigenvalue of the Hessian.
• The proof of existence relies on elementary tools from linear algebra and ODE theory, avoiding advanced PDE machinery.
• The authors use Perron’s method to establish existence and uniqueness of solutions to the Dirichlet problem for $\mathcal{P}^{+}_{1}(D^2u) + \mu u = 0$ in bounded domains.
• They establish Hölder regularity $C^{0,\frac{\beta}{N}}$ for solutions via comparison principles and barrier constructions.
• The reversed Faber-Krahn inequality is derived by comparing eigenvalues across rectangles of equal volume, showing the square yields the maximum $\mu_1^+$.

Experimental results

Research questions

• RQ1 Does the principal eigenvalue $\mu_1^+$ of the truncated Laplacian $\mathcal{P}^{+}_{1}$ increase with symmetry, specifically for rectangles of fixed volume?
• RQ2 Is strict convexity necessary for the existence of a positive eigenfunction for $\mathcal{P}^{+}_{1}$, or can such eigenfunctions exist in non-strictly convex domains like rectangles?
• RQ3 Can a reversed Faber-Krahn inequality hold for $\mathcal{P}^{+}_{1}$, where symmetric domains maximize the eigenvalue rather than minimize it?
• RQ4 How does the behavior of $\mu_1^+$ compare to the Laplacian’s principal eigenvalue $\mu(\Delta)$ in terms of subadditivity, and does Lieb’s inequality fail for $\mu_1^+$?
• RQ5 What is the regularity of solutions to the Dirichlet problem for $\mathcal{P}^{+}_{1}$, and does global Hölder continuity hold without convexity assumptions?

Key findings

• Among rectangles of fixed volume, the square maximizes the principal eigenvalue $\mu_1^+$ for the truncated Laplacian $\mathcal{P}^{+}_{1}$, establishing a reversed Faber-Krahn inequality.
• The principal eigenvalue $\mu_1^+$ for the ball of the same volume exceeds that of the square, further confirming the reversed inequality.
• An explicit eigenfunction for $\mathcal{P}^{+}_{1}$ exists in rectangles, even though they are not strictly convex, disproving the necessity of strict convexity for eigenfunction existence.
• The solution to the Dirichlet problem for $\mathcal{P}^{+}_{1}$ is globally $C^{0,\frac{\beta}{N}}$-Hölder continuous in $\overline{\Omega}$, even without assuming convexity of $\Omega$, under appropriate conditions.
• The Lieb-type subadditivity inequality $\mu(\Delta, A \cap B_x) < \mu(\Delta,A) + \mu(\Delta,B)$ does not hold for $\mu_1^+$, and in fact can be reversed for certain rectangles.
• Nonnegative supersolutions of $\mathcal{P}^{+}_{1}(D^2u) \leq f$ may fail to be globally Hölder continuous, as demonstrated by a counterexample with $u(x) = 1/(\sigma - \sum \log(\cos x_i))$ that is discontinuous in Hölder norm.