arXiv subjects
Publications and source records attributed to David Jerison.
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.
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 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.
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.
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.)
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.
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.
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.
Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. We initiate the corresponding study of higher critical points by considering a superlinear free boundary problem related to plasma confinement. The associated energy functional is nondifferentiable, and therefore standard variational methods cannot be used directly to prove the existence of critical points. Here we obtain a nontrivial generalized solution $u$ of mountain pass type as the limit of mountain pass points of a suitable sequence of $C^1$-functionals approximating the energy. We show that $u$ minimizes the energy on the associated Nehari manifold and use this fact to prove that it is nondegenerate. We use the nondegeneracy of $u$ to show that it satisfies the free boundary condition in the viscosity sense. Moreover, near any free boundary point that has a measure-theoretic normal, the free boundary is a smooth surface, and hence the free boundary condition holds in the classical sense.
Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$.
In a recent preprint, we showed that for the Dirichlet Laplacian $Δ$ on the unit disk, the wave trace ${Tr}(e^{it\sqrtΔ})$, which has complicated singularities on $2π- ε< t < 2π$, is, on the interval $2π< t < 2π+ ε$, the restriction to this interval of a $C^\infty$ function on its closure. 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.
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.
We prove a quantitative stability result for the Brunn-Minkowski inequality: if $|A|=|B|=1$, $t \in [τ,1-τ]$ with $τ>0$, and $|tA+(1-t)B|^{1/n}\leq 1+δ$ for some small $δ$, then, up to a translation, both $A$ and $B$ are quantitatively close (in terms of $δ$) to a convex set $K$.
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 show that if two components of the free boundary are close, then 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.
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 $p>N$.
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 $4$.
We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schrödinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.
We show that space-time averages of the fluctuations of internal DLA on a cylinder Z_N x Z converge in law to the Gaussian free field.