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Register a new userCompactness and dichotomy in nonlocal shape optimization
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We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two pieces whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.
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We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.
For $l$-homogeneous linear differential operators $mathcal{A}$ of constant rank, we study the implication $v_jrightharpoonup v$ in $X$ and $mathcal{A} v_jrightarrow mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)rightsquigarrow F(v)$ in $Z$, where $F$ is an $mathcal{A}$-quasiaffine function and $rightsquigarrow$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $mathscr{M}$ of measures, or $L^1$, or the Hardy space $mathscr{H}^1$; $X,, Y$ are $L^p$-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of $X,,Y,,Z$ are sharp. Analogous statements are also given in the case when $F(v)$ is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove $mathscr{H}^p$-bounds for the sequence $(F(v_j))_j$, for appropriate $p<1$, and new convergence results in the dual of Holder spaces when $(v_j)$ is $mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-beta,s}$. The choice of these Holder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
For a given Lipschitz domain $Omega$, it is a classical result that the trace space of $W^{1,p}(Omega)$ is $W^{1-1/p,p}(partialOmega)$, namely any $W^{1,p}(Omega)$ function has a well-defined $W^{1-1/p,p}(partialOmega)$ trace on its codimension-1 boundary $partialOmega$ and any $W^{1-1/p,p}(partialOmega)$ function on $partialOmega$ can be extended to a $W^{1,p}(Omega)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $mathbb{R}^dbackslashOmega$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical $W^{1-1/p,p}(partialOmega)$ space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.
This note is devoted to the study of Hyt{o}nens extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calder{o}n--Zygmund operators, fractional integrals and bilinear Calder{o}n--Zygmund operators.
For any bounded, smooth domain $Omegasubset R^2$, %(or $Omega=R^2$), we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argument for the Ginzburg-Landau type nematic liquid crystal flow.
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