Principal Dirichlet Eigenvalue Research Articles

On a front evolution problem for the multidimensional East model

We consider a natural front evolution problem the East process on $\mathbb{Z}^d, d\ge 2,$ a well studied kinetically constrained model for which the facilitation mechanism is oriented along the coordinate directions, as the equilibrium density $q$ of the facilitating vertices vanishes. Starting with a unique unconstrained vertex at the origin, let $S(t)$ consist of those vertices which became unconstrained within time $t$ and, for an arbitrary positive direction $\mathbf x,$ let $v_{\max}(\mathbf x),v_{\min}(\mathbf x )$ be the maximal/minimal velocities at which $S(t)$ grows in that direction. If $\mathbf x$ is independent of $q$, we prove that $v_{\max}(\mathbf x)= v_{\min}(\mathbf x)^{(1+o(1))}=\gamma(d) ^{(1+o(1))}$ as $q\to 0$, where $\gamma(d)$ is the spectral gap of the process on $\mathbb{Z}^d$. We also analyse the case in which some of the coordinates of $\mathbf x$ vanish as $q\to 0$. In particular, for $d=2$ we prove that if $\mathbf x$ approaches one of the two coordinate directions fast enough, then $v_{\max}(\mathbf x)= v_{\min}(\mathbf x)^{(1+o(1))}=\gamma(1) ^{(1+o(1))}=\gamma(d)^{d(1+o(1))},$ i.e. the growth of $S(t)$ close to the coordinate directions is dictated by the one dimensional process. As a result the region $S(t)$ becomes extremely elongated inside $\mathbb{Z}^d_+$. We also establish mixing time cutoff for the chain in finite boxes with minimal boundary conditions. A key ingredient of our analysis is the renormalisation technique of arXiv:1404.7257 to estimate the spectral gap of the East process. Here we extend this technique to get the main asymptotics of a suitable principal Dirichlet eigenvalue of the process.

3D positive lattice walks and spherical triangles

In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision.

Conformal Skorokhod embeddings and related extremal problems

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset \mathbb {C}$ whose exit time embeds a given probability distribution $\mu $ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $\mu $ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb {U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb {U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.

The Dirichlet problem for the logarithmic Laplacian

In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral representationwith and where Γ is the Gamma function, is the Digamma function and is the Euler Mascheroni constant. This operator arises as formal derivative of fractional Laplacians at We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of as As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

Shapes of drums with lowest base frequency under non-isotropic perimeter constraints

We study the minimizers of the sum of the principal Dirichlet eigenvalue of the negative Laplacian and the perimeter with respect to a general norm in the class of Jordan domains in the plane. This is equivalent (modulo scaling) to minimizing the said eigenvalue (or the base frequency of a drum of this shape) subject to a hard constraint on the perimeter. We show that, for all norms, a minimizer exists, is unique up to spatial translations and is convex but not necessarily smooth. We give conditions on the norm that characterize the appearance of facets and corners. We also demonstrate that near minimizers have to be close to the optimal ones in the Hausdorff distance. Our motivation for considering this class of variational problems comes from a study of random walks in random environment interacting through the boundary of their support.

Exit time moments and eigenvalue estimates

We study estimates involving the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold and L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Polya.

A Numerical study of the Dirichlet and Neumann eigenvalue problem of the Laplacian on cusp domains

Consider a planar annulus. A result of Ramm and Shivakumar states that as the inner circle moves toward the outer circle, the principal Dirichlet eigenvalue of the Laplacian decreases. Numerical experiments in that paper clearly verify this result. The purpose of this short note is to fill in a small gap in that paper: the numerical calculation of the principal eigenvalue when the two circles touch. This is a non-trivial numerical problem because the domain has a cusp which is a strong singularity. Adaptive finite element methods have difficulty converging in the presence of such singularities. Our method is to perform a transformation taking the domain to a rectangle, where it is relatively straightforward to compute the principal eigenvalue. We also calculate the minimal non-zero eigenvalue of the Neumann problem. Numerically, the Neumann eigenvalue has no such monotonicity property.

Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We study the principal Dirichlet eigenvalue of the operator $${L_A=\Delta^{\alpha/2}+Ab(x)\cdot\nabla}$$ , on a bounded C 1,1 regular domain D. Here $${\alpha\in(1,2)}$$ , $${\Delta^{\alpha/2}}$$ is the fractional Laplacian, $${A\in\mathbb{R}}$$ , and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W 1,2d/(d+α)(D). We prove that the eigenvalue remains bounded, as A→ + ∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of $${\Delta^{\alpha/2}}$$ for the Dirichlet condition.

Extremal geometry of a Brownian porous medium

The path \(W[0,t]\) of a Brownian motion on a \(d\)-dimensional torus \(\mathbb T ^d\) run for time \(t\) is a random compact subset of \(\mathbb T ^d\). We study the geometric properties of the complement \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) for \(d\ge 3\). In particular, we show that the largest regions in \(\mathbb T ^d{{\setminus }} W[0,t]\) have a linear scale \(\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}\), where \(\kappa _d\) is the capacity of the unit ball. More specifically, we identify the sets \(E\) for which \(\mathbb T ^d{{\setminus }} W[0,t]\) contains a translate of \(\varphi _d(t)E\), and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) and the \(\varepsilon \)-cover time of \(\mathbb T ^d\) as \(\varepsilon \downarrow 0\). Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in \(\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]\), where \(W_{\rho (t)}[0,t]\) is the Wiener sausage of radius \(\rho (t)\), with \(\rho (t)\) chosen much smaller than \(\varphi _d(t)\) but not too small. The idea behind this choice is that \(\mathbb T ^d {{\setminus }} W[0,t]\) consists of “lakes”, whose linear size is of order \(\varphi _d(t)\), connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of \(\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]\) as \(t\rightarrow \infty \). Our results give a complete picture of the extremal geometry of \(\mathbb T ^d{{\setminus }} W[0,t]\) and of the optimal strategy for \(W[0,t]\) to realise extreme events.

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a two-dimensional domain with L2 cells. For fixed A, and L→∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A→∞. In this case the solution equilibrates along stream lines.In this paper, we show that if bothA→∞ and L→∞, then a transition between the homogenization and averaging regimes occurs at A≈L4. When A≫L4, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A≪L4, the principal eigenvalue behaves like σ¯(A)/L2, where σ¯(A)≈AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent Lp→L∞ estimates for elliptic equations with an incompressible drift. This provides effective sub- and super-solutions for our problem.

Measuring the insulating ability of anisotropic thermal conductors via principal Dirichlet eigenvalue

We consider the thermal insulation property of homogeneous anisotropically heat-conducting bodies, i.e. those whose thermal tensor (matrix) A is constant throughout the body but not generally a constant times the identity. This anisotropy is a common feature of nano-composite materials. We propose using the principal Dirichlet eigenvalue λ of the associated elliptic differential operator −∇ ⋅ A∇ as a simple measurement for the insulating ability of the material, because the time scale of thermal flow is of the order of 1/λ. 1/λ is a generalization of the ‘R-value’ used in engineering practice as a measure of the insulating ability of isotropic conductors. If the thermal tensor A depends on parameters, e.g. inherited from nanostructure, then so does λ. It is important to know how λ depends on the parameters. For the material to be a good insulator, λ should be suppressed. But calculation or estimation of this principal elliptic eigenvalue, particularly over ranges of parameter values, is not a simple task. The focus of this paper is estimation – by fitted formulas and new exact bounds – of the principal elliptic Dirichlet eigenvalue of ellipses (2D) and ellipsoids (3D) using only simple expressions in the eigenvalues of the matrix A. Our simplest approximations and bounds avoid even the calculation of matrix eigenvalues and use in 2D merely the trace and determinant of A, and in 3D the trace and determinant of A and the trace of A2. The new bounds are shown to imply an extremal property in homogenization theory and a new condition for enhancement in Taylor dispersion. Recently, Zheng, Forest, Lipton, Zhou and Wang published formulas for the thermal tensor A in the case of strong extensional fiber-type flow of a polymer with dilute rod-like nano-inclusions, including explicitly the influence of the probability distribution of inclusion orientation. Our results are combined with these formulas to quantify the effect of variations of the probability distribution on the insulating ability of the composite.

Geometric characterization of intermittency in the parabolic Anderson model

We consider the parabolic Anderson problem $\partial_tu=\Delta u+\xi(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=\delta_0(x)$ and random i.i.d. potential $\xi$. Under the assumption that the distribution of $\xi(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to\infty$, the overwhelming contribution to the total mass $\sum_xu(t,x)$ comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field $\xi$ in a box of side length $2t\log^2t$ for which the (local) principal Dirichlet eigenvalue of the random operator $\Delta+\xi$ is close to the top of the spectrum in the box. We also prove that the shape of $\xi$ in these regions is nonrandom and that $u(t,\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.

Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\mathbb R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\mathbb R$ and $C>0$. We provide a characterization of the set of $(p,\sigma)\in\mathbb R^2$ such that the equation has no positive (super-),solutions, depending on the values of $A,B$ and the principal Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane

We prove that amongst all hyperbolic triangles of equal perimeter or quadrilaterals in a given geodesic ball the regular polygon is the unique minimum for the first Dirichlet eigenvalue. Moreover, we give a geometric description of the set of all hyperbolic triangles with a fixed base and prescribed area.

Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schrodinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process.

On the Placement of an Obstacle or a Well so as to Optimize the Fundamental Eigenvalue

We investigate how to place an obstacle B within a domain $\Omega$ in Euclidean space so as to maximize or minimize the principal Dirichlet eigenvalue for the Laplacian on $\Omega \setminus B$. The shape of B is fixed a priori (usually as a ball), and only its position varies. We establish that for a certain class of domains the minimizing B is in contact with $\partial \Omega$, while the maximizing B is in the interior, typically at the center (supposing that the domain is sufficiently symmetric for this statement to be meaningful). Under special circumstances we can characterize the optimizing configurations with multiple obstacles. Our method relies on the Hadamard perturbation formula and a moving plane analysis. Similar facts are proved when the hard obstacle is replaced by a central nonnegative potential function supported in B, and we consider the Schrödinger operator with this potential. Complementary facts are proved when the obstacle is replaced by a central nonpositive potential function.

Fluctuations of Principal Eigenvalues and Random Scales

We investigate the principal Dirichlet eigenvalue of the Laplacian with soft Poissonian obstacles in large boxes of , d≥ 2. With the help of our recent version of the method of enlargement of obstacles [18], we derive quantitative confidence intervals for these eigenvalues. We also provide less quantitative estimates, which however point out the correct size of fluctuations, and indicate a stiffness in their behavior. In the two-dimensional case we derive geometric controls, which relate these eigenvalues to certain empty circular droplets. Our results also have natural applications to the study of the location of minima of certain intermittent random variational problems, motivated by [13, 17].

Capacity and principal eigenvalues: the method of enlargement of obstacles revisited

We describe a coarse graining method, which provides lower bounds on the principal Dirichlet eigenvalue of the Laplacian in regions receiving small obstacles, and sharpens the previous method of enlargement of obstacles. Based on a quantitative Wiener criterion, one replaces the actual obstacles by obstacles of a much larger size. Controls on the shift of principal eigenvalues and capacity estimates on the locus where the Wiener criterion breaks down are derived. The results are written in a self-contained fashion.