We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to maximize the Steklov eigenvalue.
We prove existence of solutions and study properties of optimal configuration $D$. This is a free boundary problem which could be formulated as a two-sided unstable obstacle problem.
Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration $D$. On the other hand, we prove that for a convex domain $Ω$ having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration $D$ in the ball case.
This is a continuation of the paper 'Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes' by S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Again, we consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. In this paper we focus on the study of the free boundary of optimal solutions on general domains.
We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator with potential equal to a fixed multiple of the characteristic function of a subset D of omega, with measure A). We show that for minimizers, the boundary of D is analytic.
We consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $Ω$; on the other hand, for convex $Ω$ reflection symmetries are preserved. Also, we present numerical…
We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and va…
One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed to the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain $Ω\subset \mathbb R^n$, a set of very small periodically distributed se…
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szegö-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.
We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the Hölder continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain $C^{1,γ}$ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.
The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution is studied in what concerns the approximation, the regularity and the stability of the variational solutions. We extend to the evolutionary case the characterization of the Lagrange multipliers associated with the ordering constraint in terms of the characteristic functions of the coincidence sets. We give continuous dependence results, and study the asymptotic behavior as $t \to \infty$ of the solution and the coincidence sets, showing that they converge to their stationar…
A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
We provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure.
A Nonlocal $p$-Laplacian Interface Model with Sharp Interface
China Jiliang University（中国计量大学）；Tsinghua University（清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；The University of Hong Kong（香港大学）
In this paper we prove a concentration result and an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a perodic array and thermally active membranes separating the heat conductive phases.
We consider the problem of maximizing the first eigenvalue of the $p$-laplacian (possibly with non-constant coefficients) over a fixed domain $Ω$, with Dirichlet conditions along $\partialΩ$ and along a supplementary set $Σ$, which is the unknown of the optimization problem. The set $Σ$, that plays the role of a supplementary stiffening rib for a membrane $Ω$, is a compact connected set (e.g. a curve or a connected system of curves) that can be placed anywhere in $\overlineΩ$, and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This u…
In \cite{Cipriani2016}, the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study $α$-long-range divisible sandpiles, similar to those introduced in \cite{Frometa2018}. We show that, for $α\in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $α/2$. However, for $α\in [2,\infty)$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane mode…
We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neig…
We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-Δ)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-Δ)^su(t,x)$ and p…
