Search

We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by striking results of T. Colding and W. Minicozzi concerning finitely connected, embedded, minimal surfaces. We review our earlier work on the simplest case, one-phase free boundaries in the plane in which the positive phase is simply connected. We also prove a new, purely topological, effective removable singularities theorem for free boundaries. At the same time, we formulate some open problems concerning the multiply connected case and make connections with the theory of minimal surfaces and semilinear variational problems.

Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.

International audience

We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the rate at which the fixed boundary converges to its limit cone at each boundary point., Comment: 34 pages

The Friedland–Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here, we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions., National Science Foundation (U.S.) (DMS 1069225), National Science Foundation (U.S.) (DMS 1500771), Stefan Bergman Trust

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to $Lu=1$, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, $1/u$, acts as an effective potential. Hence from the single measurement of $u$, we obtain, via $1/u$, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum. (This version strengthens and simplifies the results of the first one by replacing a global bi-Lipschitz hypothesis on the domain with a local bi-Lipschitz hypothesis. It improves on the second version by adding pictures and numerical examples. This version is identical to the third version; all that is changed is to correct some tex mistakes in symbols in this abstract. There are no changes to the paper itself.), Comment: 33 pages

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schr\"odinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schr\"odinger eigenfunctions for periodic and other ordered potentials, for many forms of disordered potentials the eigenfunctions remain essentially localized in a very small subset of the initial domain. A celebrated example is Anderson localization, for which, in a continuous version, the potential is a piecewise constant function on a uniform grid whose values are sampled independently from a uniform random distribution. We present here a new method for approximating the eigenvalues and the subregions which support such localized eigenfunctions. This approach is based on the recent theoretical tools of the localization landscape and effective potential. The approach is deterministic, predicting quantities that depend sensitively on the particular realization, rather than furnishing statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. These methods, which have only been partially justified theoretically, enable the calculation of the locations and shapes of the approximate supports of the eigenfunctions, the approximate values of many of the eigenvalues, and of the eigenvalue counting function and density of states, all at the cost of solving a single source problem for the same elliptic operator. We study the effectiveness and limitations of the approach through extensive computations in one and two dimensions, using a variety of piecewise constant potentials with values sampled from various different correlated or uncorrelated random distributions., Comment: 24 pages, 19 figures

We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the {\it Hots Spots Conjecture} of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar\'e inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible., Comment: 22 pages, this new version corrects one word in the introduction, Aguilera is the name of the first author of a paper cited (not Athanosopoulos). Thanks to Joel Spruck for pointing out this error

We prove a quantitative stability result for the Brunn–Minkowski inequality: if | A | = | B | = 1 , t ∈ [ τ , 1 − τ ] with τ > 0 , and | t A + ( 1 − t ) B | 1 / n ≤ 1 + δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K .

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of 1D systems, periodic or random.

Let each of [superscript n] particles starting at the origin in Z[superscript 2] perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of [superscript n] occupied sites is (with high probability) close to a disk B [subscript r] of radius r=√n/pi. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant [superscript C] such that with probability [superscript 1], B [subscript r - C log r] C A(pi r[superscript 2]) C B [subscript r+ C log r] for all sufficiently large r., National Science Foundation (U.S.) (grant DMS-1069225), National Science Foundation (U.S.) (grant DMS-0645585), National Science Foundation (U.S.) (Postdoctoral Research Fellowship)

This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that if the Lipschitz constant of the initial free boundary in a unit ball is small, then for small uniform positive time the solution is smooth. This result improves on our earlier results in (4) because it is scale- invariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz ini- tial free boundary.

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex. Our proof relies on measuring the perimeter of a set through a Hilbertian fractional Sobolev norm, a fact that we believe has its own interest., National Science Foundation (U.S.) (Grant DMS-1069225)

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random., Comment: 5 pages, 3 figures

In a recent preprint, we showed that for the Dirichlet Laplacian Δ on the unit disk, the wave trace $$Tr\left( {{e^{it\sqrt \Delta }}} \right)$$ , which has complicated singularities on 2π−e < t < 2π, is bounded and infinitely differentiable as t →2π from the right. In this paper, we prove the analogue of this somewhat counter-intuitive result for the Friedlander model. The proof for the Friedlander model is simpler and more transparent than in the case of the unit disk and suggests the direction to follow to treat the case of general smooth strictly convex domains.

We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply connected. We characterize the way in which the curvature of the free boundary can tend to infinity. Indeed, if curvature tends to infinity, then two components of the free boundary are close, and the solution locally resembles an entire solution discovered by Hauswirth, Hélein, and Pacard, whose free boundary has the shape of a double hairpin. Our results are analogous to theorems of Colding and Minicozzi characterizing embedded minimal annuli, and a direct connection between our theorems and theirs can be made using a correspondence due to Traizet., National Science Foundation (U.S.) (Grant DMS-1069225)

We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension 4. Keywords: Free Boundary; Minimal Surface; Free Boundary Problem; Stable Cone; Strict Subsolution

We prove the absence of positive eigenvalues of Schrodinger operators $ H=-\Delta+V $ on Euclidean spaces $ \mathbb{R}^n $ for a certain class of rough potentials $V$. To describe our class of potentials fix an exponent $q\in[n/2,\infty]$ (or $q\in(1,\infty]$, if $n=2$) and let $\beta(q)=(2q-n)/(2q)$. For the potential $V$ we assume that $V\in L^{n/2}_{{\rm{loc}}}(\mathbb{R}^n)$ (or $V\in L^{r}_{{\rm{loc}}}(\mathbb{R}^n)$, $r>1$, if $n=2$) and $\begin{equation*}$ $\lim_{R\to\infty}R^{\beta(q)}||V||_{L^q(R\leq |x|\leq 2R)}=0\,.$ $\end{equation*}$ Under these assumptions we prove that the operator $H$ does not admit positive eigenvalues. The case $q=\infty$ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form $\begin{equation*}$ $||W_m u||_{l^a(L^{p’(q)})(\mathbb R^n)}\leq C_q||W_m|x|^{\beta(q)}(\Delta+1)u||_{l^a(L^{p(q)})(\mathbb{R}^n)}$ $\end{equation*}$ for all smooth compactly supported functions $u$ and a suitable sequence of weights $W_m$, where $p(q)$ and $p’(q)$ are dual exponents with the property that $1/p(q)-1/p’(q)=1/q$.

This chapter discusses the continuum limit of internal Diffusion-Limited Aggregation (DLA), a random lattice growth model governed by a deterministic fluid flow equation known as Hele-Shaw flow. The internal DLA model was introduced in 1986 by Meakin and Deutch to describe chemical processes such as electropolishing, etching, and corrosion. The chapter focuses primarily on fluctuations, and seeks to prove the analogous results for the lattice cylinder. In the case of the cylinder, the fluctuations are described in terms of the Gaussian Free Field exactly. The main tools used in the proofs are martingales. As the chapter shows, the martingale property in this context is the counterpart in probability theory of well-known conservation laws for Hele-Shaw flow.

We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$. In particular, inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space of fractional order $B(2-\frac{2}{N},N)$, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-\frac{2}{N},N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ of codimension $2$. The most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-\frac{2}{N},p)$ with $p>N$., 26 pages

For solutions of semilinear elliptic equations, we show the link that exists between global minimizers on R n−1 and the existence of nontrivial monotonic solutions on R n . We prove that the existence of a symmetric global minimizer on R n−1 implies the existence of a nonplanar monotonic solution on R n which is a global minimizer. Moreover, we prove that every monotonic solution on R n is a global minimizer if and only if its up and down limits are global minimizers.

Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We deduce J. Rauch's "hot spots" conjecture in the following form. If the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. In fact the maximum point reaches the boundary in finite time if the boundary has positive curvature. Results of this type have already been proved by Bafiuelos and Burdzy [BB] using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions. We introduce here a new technique based on deformation of the domain. An advantage of our method is that it works even in the case of multiple eigenvalues. On the way toward our results, we prove monotonicity properties for Neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the Dirichlet problem of independent interest.

The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance comparable to the inradius, uniformly for arbitrarily large diameter. In addition, we will estimate the location of other level sets of u by showing that u is well-approximated by the first eigenfunction of a naturally associated ordinary differential (Schrodinger) operator. We intend to show in a separate paper that the estimates here are best possible in order of magnitude. The present paper depends on the ideas and results of our earlier work [J] and [GJ], where detailed estimates for the zero set of the second eigenfunction (or first nodal line) are obtained. The paper [J] also contains some estimates for the first eigenfunction and lowest eigenvalue, but the techniques of [GJ] and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here. The maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency. The first nodal line is the stationary curve of the drum at the second resonant frequency. The maximum is harder to find experimentally than the nodal line because it is a single point. Its location has less influence on the eigenvalue or Dirichlet integral, so it is also harder to locate mathematically. Another way to describe the difficulty is as follows. To find the maximum of the first eigenfunction we will need to estimate its first directional derivative. Derivatives of the first eigenfunction are, roughly speaking, analogous to the second eigenfunction because they are solutions to an eigenfunction equation. Moreover, convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components. But the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary.

Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d > 2. We show that A(t) is approximately spherical, up to an O(\sqrt{\log t}) error., 17 pages, minor revision

A smooth demand function is generated by utility maximization if and only if its Slutsky matrix is symmetric and negative semidefinite. Slutsky symmetry is equivalent to absence of smooth revealed preference cycles, cf. Hurwicz and Richter (Econometrica 1979). To observe such a cycle would require a continuum of data. We characterize Slutsky symmetry by means of discrete “antisymmetric” revealed preference cycles consisting of either three or four observations.

In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on Z[superscript d] with t particles is almost surely spherical up to a maximal error of O(logt) if d=2 and O(logt√) if d≥3. This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r[superscript 1−d/2] (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field., National Science Foundation (U.S.) (grant DMS-0645585), National Science Foundation (U.S.) (grant DMS-1105960), National Science Foundation (U.S.) (Postdoctoral Research Fellowship), National Science Foundation (U.S.) (grant DMS-1069225)

We define measures of violations of Slutsky symmetry and negative semidefiniteness and relate them to measures of revealed preference inconsistencies exhibited by nonoptimizing demand behavior. The degree of Slutsky asymmetry is shown to restrict the rate at which real income can rise everywhere along smooth loops in income and price space. The largest eigenvalue of the quadratic form of the Slutsky matrix is used to bound violations of the weak axiom. The sizes of the violations of either Slutsky condition are used to bound the distance between the given demand function and approximating functions that satisfy that Slutsky condition exactly.

We prove an analogue for a one-phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy-minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound., National Science Foundation (U.S.) (grant DMS-0244991)

Small deviations from optimizing behavior can have substantial effects on economic equilibria. This paper analyzes deviations from optimization in consumer demand. We show how a measure of the Slutsky asymmetry of a demand function is related to the rate at which real income can rise along a smooth revealed preference cycle. We also relate the Slutsky asymmetry measure to the minimum distance between the given demand function and a function that is derivable from utility maximization. The results yield simple quantitative revealed preference interpretations for violations of Slutsky symmetry while suggesting how to compare the sizes of revealed preference inconsistencies.

The purpose of this paper was to extend some of the results described in [12]. Some sharp estimates are proven for the wave operator on compact manifolds for two or more dimensions. The equations include the natural discrete analog of Strichartz's restriction theorem for the light cone. 5 refs.

We consider a family of domains $(\Omega_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $\Omega_0 = \{(x,y): 0, Comment: 19 pages, 2 figures

In this article we analyze viscosity solutions of the one phase Hele-Shaw problem in the plane and the corresponding free boundaries near a singularity. We find, up to order of magnitude, the speed at which the free boundary moves starting from a wedge, cusp, or finger-type singularity. Maximum principle-type arguments play a key role in the analysis.

Suppose that Ω and Ω1 are convex, open subsets of Rn. Denote their convex combination by The Brunn-Minkowski inequality says that (vol Ω)t≥ (1 -t) vol Ω0 1/N +t Vol Ω for 0≤t ≤ l. Moreover, if there is equality for some t other than an endpoint, then the domains Ω1 and Ω0 are translates and dilates of each other. Borell proved an analogue of the Brunn—Minkowski inequality with capacity (defined below) in place of volume. Borel’s theorem [B] says THEOREM A. Let Ωt= tΩ1+ (1—t)Ω0 be a convex combination of two convex subsets of RN,N≥3. Then cap The main purpose of this note is to prove.