Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHigher integrability for measures satisfying a PDE constraint
49
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We establish higher integrability estimates for constant-coefficient systems of linear PDEs [ mathcal A mu = sigma, ] where $mu in mathcal M(Omega;V)$ and $sigmain mathcal M(Omega;W)$ are vector measures and the polar $frac{dmu}{d |mu|}$ is uniformly close to a subspace $L$ of $V$ intersecting the wave cone of $mathcal A$ only at the origin. More precisely, we prove local compensated compactness estimates of the form [ |mu|_{L^p(Omega)} lesssim |mu|(Omega) + |sigma|(Omega), qquad Omega Subset Omega. ] Here, the exponent $p$ belongs to the (optimal) range $1 leq p < d/(d-k)$, $d$ is the dimension of $Omega$, and $k$ is the order of $mathcal A$. For canceling constant-rank operators we also obtain the limiting case $p = d/(d-k)$. We consider applications to compensated compactness as well as applications to the theory of functions of bounded variation and bounded deformation.
rate research
Read More
We give a simple criterion on the set of probability tangent measures $mathrm{Tan}(mu,x)$ of a positive Radon measure $mu$, which yields lower bounds on the Hausdorff dimension of $mu$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is also discussed for such measures.
In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) abla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse H{o}lders regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $alpha$ in the H{o}lder $C^alpha$-regularity theory on the textup{BMO}-semi norm of the skew-symmetric part of $mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $mathbf{A}$ is assumed to be zero.
In this expository article we introduce a diagrammatic scheme to represent reverse classes of weights and some of their properties.
This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stoppi
In this survey we present some recent results concerning computer-assisted proofs in partial differential equations, focusing in those coming from problems in incompressible fluids. Particular emphasis is put on the techniques, as opposed to the results themselves.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
