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Register a new userCompensated compactness: continuity in optimal weak topologies
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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.
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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.
Concentration-compactness is used to prove compactness of maximising sequences for a variational problem governing symmetric steady vortex-pairs in a uniform planar ideal fluid flow, where the kinetic energy is to be maximised and the constraint set comprises the set of all equimeasurable rearrangements of a given function (representing vorticity) that have prescribed impulse (lnear momentum). A form of orbital stability is deduced.
The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Omega)$ compactly embeds in $W^{-1,q}(Omega)$, for every $q < 2$ and for any open and bounded set $Omega$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities.
We study the weak continuity of two interrelated non-linear partial differential equations, the Yang-Mills equations and the Gau{ss}-Codazzi-Ricci equations, involving $L^p$-integrable connections. Our key finding is that underlying cancellations in the curvature form, especially the div-curl structure inherent in both equations, are sufficient to pass to the limit in the non-linear terms. We first establish the weak continuity of Yang-Mills equations and prove that any weakly converging sequence of weak Yang-Mills connections in $L^p$ converges to a weak Yang-Mills connection. We then prove that, for a sequence of isometric immersions with uniformly bounded second fundamental forms in $L^p$, the curvatures are weakly continuous, which leads to the weak continuity of the Gau{ss}-Codazzi-Ricci equations with respect to sequences of isometric immersions with uniformly bounded second fundamental forms in $L^p$. Our methods are independent of dimensions and do not rely on gauge changes.
We develop a functional framework suitable for the treatment of partial differential equations and variational problems posed on evolving families of Banach spaces. We propose a definition for the weak time derivative which does not rely on the availability of an inner product or Hilbertian structure and explore conditions under which the spaces of weakly differentiable functions (with values in an evolving Banach space) relate to the classical Sobolev--Bochner spaces. An Aubin--Lions compactness result in this setting is also proved. We then analyse several concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev--Bochner spaces. We conclude with the formulation and proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising in particular the evolutionary $p$-Laplace equation on a moving domain or surface) and identify some additional evolutionary problems that can be appropriately formulated with the abstract setting developed in this work.
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