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Ryan Gibara

Publications and source records attributed to Ryan Gibara.

At least 19 recordsLinked to original sources
arXiv2025

Sharp H\"older regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces

We prove global H\"older regularity result for weak solutions $u\in N^{1,p}(\Omega, \mu)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overline\nu, \] where $1<p<\infty$ and $\overline\nu \in (N^{1,p}(\Omega,\mu))^*$ is a signed Radon measure supported in $\overline \Omega$. Here, $\Omega$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincar\'e inequality, and $\nabla u$ is the Cheeger gradient. The regularity results obtained in this paper improve on earlier estimates proved by the authors in \cite{CGKS} for the study of the Neumann problem, and have applications to the regularity of solutions of nonlocal PDE in doubling metric spaces. Moreover, the obtained H\"older exponent matches with the known sharp result in the Euclidean case \cite{CSt,BLS,BT}.

math.AP↗
arXiv2024

Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-\Delta_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,\nu)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{\v{c}}anov and Ostrovski{\u{i}} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincar\'e inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, \mu)$ that arises as an hyperbolic filling of $Z$.

math.AP↗
arXiv2024

On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure $\mu$ with $\mu(X) = \infty$ and $0 < \mu(B(x, r)) < \infty$ for all $x \in X$ and $r \in (0, \infty)$ Our objective is to understand the relationship between the Dirichlet space $D^{1,p}(X)$, defined using upper gradients, and the Newton-Sobolev space $N^{1,p}(X)+\mathbb{R}$, for $1\le p<\infty$. We show that when $X$ is of uniformly locally $p$-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space $\mathbb{H}^n$ with $n\ge 2$, these two spaces coincide precisely when $1\le p\le n-1$. We also provide additional characterizations of when a function in $D^{1,p}(X)$ is in $N^{1,p}(X)+\mathbb{R}$ in the case that the two spaces do not coincide.

math.MG↗
arXiv2024

On the Dirichlet-to-Neumann Map for the $p$-Laplacian on a Metric Measure Space

In this note, we construct a Dirichlet-to-Neumann map, from a Besov space of functions, to the dual of this class. The Besov spaces are of functions on the boundary of a bounded, locally compact uniform domain equipped with a doubling measure supporting a $p$-Poincar\'e inequality so that this boundary is also equipped with a Radon measure that has a codimensional relationship with the measure on the domain. We construct this map via the following recipe. We show first that solutions to Dirichlet problem for the $p$-Laplacian on the domain with prescribed boundary data in the Besov space induce an operator that lives in the dual of the Besov space. Conversely, we show that there is a solution, in the homogeneous Newton-Sobolev space, to the Neumann problem for the $p$-Laplacian with the Neumann boundary data given by a continuous linear functional belonging to the dual of the Besov space. We also obtain bounds on its operator norm in terms of the norms of trace and extension operators that relate Newton-Sobolev functions on the domain to Besov functions on the boundary.

math.AP↗
arXiv2023

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

We give a sharp Hausdorff content estimate for the size of the accessible boundary of any domain in a metric measure space of controlled geometry, i.e., a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality for a fixed $1\le p<\infty$. This answers a question posed by Jonas Azzam. In the process, we extend the result to every doubling gauge in metric measure spaces which satisfies a codimension one bound.

math.MG↗
arXiv2023

Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincar\'e inequality

We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, "visibility" indicates the existence of John curves connecting the interior point to the points on the "visible boundary". In this paper, we provide an affirmative answer in the setting of a doubling metric measure space supporting a $p$-Poincar\'e inequality for $1<p<\infty$, thus extending the results of [20,2,9] to non-Ahlfors regular spaces. We show that $t$-codimensional thickness of the boundary for $0<t<p$ implies $p$-codimensional thickness of the visible boundary. For such domains we prove that traces of Sobolev functions on the domain belong to the Besov class of the visible boundary.

math.MG↗
arXiv2023

Solving a Dirichlet problem for unbounded domains via a conformal transformation

In this paper, we solve the $p$-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter $p$. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.

math.AP↗
arXiv2023

Fractional maximal functions and mean oscillation on bounded doubling metric measure spaces

Let $(X,d,μ)$ be a doubling metric measure space. We consider the behaviour of the fractional maximal function $M^α$ for $0\leq α 0$, we additionally assume that the space is bounded. We show that $M^α$ is bounded from $BMO$ to $BLO$, a subclass of $BMO$, and maps $VMO$ to itself when $μ$ has the annular decay property. We also show by means of examples that the action of $M^α$ is not continuous on these function spaces.

math.FA↗
arXiv2022

Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condition with respect to $μ$ for some $0<θ<p$, then there is a bounded linear trace operator $T:D^{1,p}(\Om)\rightarrow HB^{1-θ/p}(\partial\Om)$ and a bounded linear extension operator $E:HB^{1-θ/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$ that is a right-inverse of $T$.

math.FA↗
arXiv2022

Conformal transformation of uniform domains under weights that depend on distance to the boundary

The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.

math.MG↗
arXiv2022

On the vanishing of Green's function, desingularization and Carleman's method

The subject of the present paper is the phenomenon of vanishing of the Green function of the operator $-Δ+ V$ on $\mathbb R^3$ at the points where a potential $V$ has positive critical singularities. More precisely, imposing minimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upper bound on the order of vanishing of the Green function. As a by-product of our proof, we improve the existing results on the strong unique continuation for eigenfunctions of $-Δ+ V$ in dimension $d=3$.

math.AP↗
arXiv2022

Vanishing Mean Oscillation and Continuity of Rearrangements

We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping f -> f* preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity holds at points in VMO (under certain conditions). This also applies to the symmetric decreasing rearrangement. Many examples are included to illustrate the results.

math.FA↗
arXiv2021

Accessible parts of the boundary for domains in metric measure spaces

We prove in the setting of $Q$--Ahlfors regular PI--spaces the following result: if a domain has uniformly large boundary when measured with respect to the $s$--dimensional Hausdorff content, then its visible boundary has large $t$--dimensional Hausdorff content for every $0<t<s\leq Q-1$. The visible boundary is the set of points that can be reached by a John curve from a fixed point $z_{0}\in Ω$. This generalizes recent results by Koskela-Nandi-Nicolau (from $\mathbb{R}^2$) and Azzam ($\mathbb{R}^n$). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

math.MG↗
arXiv2021

Mean oscillation bounds on rearrangements

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on $\mathbb{R}^n$. For the decreasing rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett--DeVore--Sharpley, $\|f^*\|_{BMO(\mathbb{R}_+)}\leq C_n \|f\|_{BMO(\mathbb{R}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt{n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón--Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.

math.FA↗
arXiv2020

BMO and the John-Nirenberg Inequality on Measure Spaces

We study the space BMO in the general setting of a measure space $\mathbb{X}$ with a fixed collection $\mathscr{G}$ of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in $\mathscr{G}$. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space $\mathbb{X}$ for BMO to be a Banach space modulo constants. We also develop the notion of a Denjoy family $\mathscr{G}$, which guarantees that functions in BMO satisfy the John-Nirenberg inequality on the elements of $\mathscr{G}$.

math.FA↗
arXiv2020

Geometric maximal operators and BMO on product bases

We consider the problem of the boundedness of maximal operators on BMO on shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO.

math.FA↗
arXiv2019

Potential envelope theory and the local energy theorem

We consider a one--particle bound quantum mechanical system governed by a Schrödinger operator $\mathscr{H} = -Δ+ v\,f(r)$, where $f(r)$ is an attractive central potential, and $v>0$ is a coupling parameter. If $ϕ\in \mathcal{D}(\mathscr{H})$ is a `trial function', the local energy theorem tells us that the discrete energies of $\mathscr{H}$ are bounded by the extreme values of $(\mathscr{H}ϕ)/ϕ,$ as a function of $r$. We suppose that $f(r)$ is a smooth transformation of the form $f = g(h)$, where $g$ is monotone increasing with definite convexity and $h(r)$ is a potential for which the eigenvalues $H_n(u)$ of the operator $\mathcal{H}=-Δ+ u\, h(r)$, for appropriate $u >0$, are known. It is shown that the eigenfunctions of $\mathcal{H}$ provide local-energy trial functions $ϕ$ which necessarily lead to finite eigenvalue approximations that are either lower or upper bounds. This is used to extend the local energy theorem to the case of upper bounds for the excited-state energies when the trial function is chosen to be an eigenfunction of such an operator $\mathcal{H}$. Moreover, we prove that the local-energy approximations obtained are identical to `envelope bounds', which can be obtained directly from the spectral data $H_n(u)$ without explicit reference to the trial wave functions.

math-ph↗