arXiv subjects

Daniel E. Spector

Publications and source records attributed to Daniel E. Spector.

3 recordsLinked to original sources

arXiv2020

Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers

In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $Ω\subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$, the space of functions of Bounded Mean Oscillation of John & Nirenberg. A version of Taylor's theorem is first shown to be valid provided the integrand $W$ has polynomial growth. This result is then used to demonstrate that, for the Dirichlet, Neumann, and mixed problems, every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the second variation of the energy is uniformly positive is a strict local minimizer of the energy in $W^{1,BMO}(Ω;\mathbb{R}^N)$, the subspace of the Sobolev space $W^{1,1}(Ω;\mathbb{R}^N)$ for which the weak derivative $\nabla\mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$.

math.AP↗

arXiv2020

BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension

In this manuscript two $BMO$ estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the $BMO$-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the $BMO$-seminorm of the symmetric part of its gradient, that is, a Korn inequality in $BMO$. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such an equilibrium solution, then there is a $BMO$-neighborhood in strain space where there are no other equilibrium solutions.

math.AP↗

arXiv2015

$L^p$-Taylor approximations characterize the Sobolev space $W^{1,p}$

In this note, we introduce a variant of Calderón and Zygmund's notion of $L^p$-differentiability - an \emph{$L^p$-Taylor approximation}. Our first result is that functions in the Sobolev space $W^{1,p}(\mathbb{R}^N)$ possess a first order $L^p$-Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the $L^p$-differentiability of Sobolev functions. In fact, the main result we announce here is that the first order $L^p$-Taylor approximation characterizes the Sobolev space $W^{1,p}(\mathbb{R}^N)$, and therefore implies $L^p$-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderón-Zygmund classes with others due to Bourgain, Brezis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising, however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.

math.FA↗