arXiv subjects
Publications and source records attributed to Steve Hofmann.
We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain $Ω$ with Ahlfors regular boundary, and we show that the $A_\infty$ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher.
Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partialΩ$ is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $\partialΩ$ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $\partialΩ$ can be covered $\mathcal{H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $Ω$ and to the fact that $\partialΩ$ possesses exterior corkscrew points in a qualitative way $\mathcal{H}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.
In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well-posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for a higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.
We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of $u$ exist and lie in a Lebesgue space $L^p(\mathbb{R}^n)$ or Sobolev space $\dot W^p_{\pm 1}(\mathbb{R}^n)$. Even in the case where $u$ is a solution to a second order equation, our results are new for certain values of~$p$.
We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $Ω$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial Ω$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $Ω$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $Ω$ satisfies an interior Corkscrew condition, in the special case that $L$ is the Laplacian.
In the present paper we prove that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.
We present an alternative proof of a result of Kenig and Toro, which states that if $Ω\subset \mathbb{R}^{n+1}$ is a two sided NTA domain, with Ahlfors-David regular boundary, and the $\log$ of the Poisson kernel associated to $Ω$ as well as the $\log$ of the Poisson kernel associated to ${Ω_{\rm ext}}$ are in VMO, then the outer unit normal $ν$ is in VMO . Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that $\partialΩ$ is uniformly rectifiable, and that $\partialΩ$ coincides with the measure theoretic boundary of $Ω$ a.e. with respect to Hausdorff $H^n$ measure.
In the present paper we sketch the proof of the fact that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.
In this paper we establish square-function estimates on the double and single layer potentials for divergence-form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space. This generalizes known results for variable-coefficient second-order operators, and also for constant-coefficient higher-order operators.
In the present paper we prove that for any open connected set $Ω\subset{\mathbb R}^{n+1}$, $n\geq 1$, and any $E\subset \partialΩ$ with $0<{\mathcal H}^n(E)<\infty$ absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.
We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $Ω\subset \mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant $L^q$ bounds, with $q>1$, on the Poisson kernel.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew condition, and whose connected components are all chord-arc domains (with uniform control of the various constants). As a consequence, we deduce that $E$ has big pieces of sets for which harmonic measure belongs to weak-$A_\infty$
Assuming $T_0$ to be an m-accretive operator in the complex Hilbert space $\mathcal{H}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T = T_0 + W$ and prove stability of square root domains, that is, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for PDE applications. We apply this approach to elliptic second-order partial differential operators of the form $$ - div(a\nabla \, \cdot \,) + \big(\mathbf{B}_1\cdot \nabla \cdot \big) + div \big(\mathbf{B}_2 \cdot \big) + V $$ in $L^2(Ω)$ on certain open sets $Ω\subseteq \mathbb{R}^n$, $n \in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial Ω$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\partialΩ$.
We determine square root domains for non-self-adjoint Sturm--Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb{R}$, the half-line $(a,\infty)$, $a \in \mathbb{R}$, or with the bounded interval $(a,b) \subset \mathbb{R}$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e.\ on $(a,b)$, we treat all separated boundary conditions at $a, b$.)
We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,Ω,Γ} = -\nabla\cdot a \nabla$ in $L^2(Ω)$, with mixed boundary conditions on $\partial Ω$, with respect to additive perturbations. We consider open, bounded, and connected sets $Ω\in \mathbb{R}^n$, $n \in \mathbb{N} \backslash\{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,Ω,Γ}$ with respect to additive potential perturbations $V \in L^p(Ω) + L^{\infty}(Ω)$, $p>n/2$. Special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $Ω:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.