Brunn-Minkowski Inequality for p-Harmonic Measures
We prove a local Brunn-Minkowski inequality for a functional corresponding to p-harmonic measures for 2 < p < n+1.
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arXiv subjects
Murat Akman
Publications and source records attributed to Murat Akman.
On the Minkowski problem for p-harmonic measures
We study the Minkowski problem corresponding to the p-harmonic measures and obtain results previously known for harmonic measures due to Jerison. We show that a class of Borel measures on spheres can be prescribed by p-harmonic measures on convex domains.
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Borderline gradient continuity for the normalized $p$-parabolic operator
In this paper, we prove gradient continuity estimates for viscosity solutions to $Δ_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $Δ_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain Hölder continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves the gradient Hölder continuity result in [3] which considers bounded $f$. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.
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Failure of Fatou type theorems for solutions to PDE of $p$-Laplace type in domains with flat boundaries
Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive integer let $ Λ_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3, $ be a $k$-dimensional plane with $ 0 \in Λ_k.$ If $n-k < p <\infty$, we first study the Martin boundary problem for solutions to the $p$-Laplace equation (called $p$-harmonic functions) in $ \mathbb{R}^{n} \setminus Λ_k $ relative to $ \{0\}. $ We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for $p$-harmonic functions in $ \mathbb{R}^{2}_+ $ to $p$-harmonic functions in $ \mathbb{R}^{n} \setminus Λ_k $ when $ n-k < p <\infty$. Finally, we discuss generalizations of our work to solutions of $ p $-Laplace type PDE (called $ \mathcal{A}$-harmonic functions).
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Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition
Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative versions of openness and path-connectedness. Assume that $Ω$ satisfies the so-called capacity density condition. Let $L_0u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real (non-necessarily symmetric) uniformly elliptic operators, and write $ω_{L_0}$, $ω_L$ for the associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $ω_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $ω_{L_0}$. We show that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to $ω_{L_0}$, then $ω_L\in A_\infty(ω_{L_0})$. Moreover, $ω_L\in RH_q(ω_{L_0})$ for any given $1<q<\infty$ if the Carleson measure condition is assumed to hold with a sufficiently small constant. This extends previous work of Fefferman-Kenig-Pipher and Milakis-Pipher-Toro who considered Lipschitz and chord-arc domains. Here we go beyond as the capacity density condition is much weaker than the existence of exterior Corkscrew balls. The "large constant" case, where the discrepancy satisfies a Carleson measure condition, is new even for nice domains such as the unit ball, the upper half-space, or Lipschitz domains, and is obtained using the method of extrapolation of Carleson measure. Our domains do not have a nice surface measure: all the analysis is done with the underlying measure $ω_{L_0}$. When particularized to Lipschitz, chord-arc, or 1-sided chord-arc domains, we recover previous results and extend some of them. Our arguments rely on the square function and non-tangential estimates proved in arXiv:2103.10046.
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Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition
Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $Ω$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that $Ω$ satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider $L_0 u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$, two real (non-necessarily symmetric) uniformly elliptic operators in $Ω$, and write $ω_{L_0}$, $ω_L$ for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $ω_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $ω_{L_0}$. In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in arXiv:1901.08261.
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On a Theorem of Wolff Revisited
We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $ \mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{iθ} )$, which is $ 2π/N $ periodic in the $ θ$ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $ c^{-1} \leq \int_{-π}^π V ( e^{iθ} ) d θ\leq c$. In case $2<p<\infty $ we give a more or less explicit example of $V$ and our work is an extension of a result of Wolff on $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$. Using our first result, we extend the work of Wolff on failure of Fatou type theorems for $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$ for $p$-harmonic functions, $1< p\neq 2<\infty$. Finally, we also outline the modifications needed for extending the work of Llorente, Manfredi, and Wu regarding failure of subadditivity of $p$-harmonic measure on $ \partial \mathbb{R}^{2}_+ $ to $\partial B (0, 1)$.
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Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $\mathbb{R}$^n
We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 \cos α\, | x| \} \quad \mbox{for fixed}\, \, α\in (0, π], \] with continuous boundary value zero on $ \partial K ( α) \setminus \{0\}$. In our main result we show that if $ u $ has continuous boundary value $0$ on $ \partial K ( π)$ then $u$ is homogeneous of degree $ 1 - (n-1)/p $ when $ p > n - 1. $ Applications of this result are given to a Minkowski type regularity problem in $ \mathbb{R}^{n}$ when $n=2,3$.
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On a Bernoulli-type overdetermined free boundary problem
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1 0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on $\partialΩ$, such that $|\nabla u|=c$ on $\partial Ω$. Moreover, $\partialΩ$ is $C^{1,γ}$ for some $γ>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n \setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$.
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Rectifiability, interior approximation and Harmonic Measure
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $H^n$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $Ω\subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $H^n|_{\partialΩ}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $Ω$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results by Azzam-Hofmann-Martell-Mayboroda-Mourgoglou-Tolsa-Volberg, we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure.
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The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity
In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\mbox{Cap}_{\mathcal{A}}$, where $\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the $p$-Laplace equation and whose solutions in an open set are called $ \mathcal{A}$-harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ \left[\mbox{Cap}_\mathcal{A}(λE_1 +(1-λ)E_2)\right]^{\frac{1}{(n-p)}}\geqλ\left[\mbox{Cap}_\mathcal{A}(E_1)\right]^{\frac{1}{(n-p)}}+(1-λ)\left[\mbox{Cap}_\mathcal{A}(E_2 )\right]^{\frac{1}{(n-p)}} \] when $1<p<n$, $0<λ<1$, and $E_1, E_2$ are convex compact sets with positive $\mathcal{A}$-capacity. Moreover, if equality holds in the above inequality for some $E_1$ and $E_2, $ then under certain regularity and structural assumptions on $\mathcal{A}$, we show that these two sets are homothetic. In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set $E$ with nonempty interior and its $\mathcal{A}$-harmonic capacitary function in the complement of $E$. If $μ_E$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure $μ$ on $\mathbb{S}^{n-1}$, find necessary and sufficient conditions for which there exists $E$ as above with $μ_E =μ$. We show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem for volume as well as in the work of Jerison for electrostatic capacity. Using the Brunn-Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation when $p\neq n- 1$ and translation and dilation when $p = n-1$.
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The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green's function
In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\leq n\leq p<\infty$. For a convex compact set $E$ in $\mathbb{R}^{n}$, we define and then prove the existence and uniqueness of the so called $\mathcal{A}$-harmonic Green's function for the complement of $E$ with pole at infinity. We then define a quantity $\mbox{C}_{\mathcal{A}}(E)$ which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that $\mbox{C}_{\mathcal{A}}(\cdot)$ satisfies the following Brunn-Minkowski type inequality \[ \left[\mbox{C}_\mathcal{A} ( λE_1 + (1-λ) E_2 )\right]^{\frac{1}{p-n}} \geq λ\, \left[\mbox{C}_\mathcal{A} ( E_1 )\right]^{\frac{1}{p-n}} + (1-λ) \left[\mbox{C}_\mathcal{A} (E_2 )\right]^{\frac{1}{p-n}} \] when $n<p<\infty$, $0 \leq λ\leq 1$, and $E_1, E_2$ are nonempty convex compact sets in $\mathbb{R}^{n}$. We also show that $\mbox{C}_\mathcal{A}(\cdot)$ satisfies a similar inequality when $p=n$. Moreover, if equality holds in the either of these inequalities for some $E_1$ and $E_2$ then under certain regularity and structural assumptions on $\mathcal{A}$ we show that these two sets are homothetic. In the second part of this article we study a Minkowski type problem for a measure associated to the $\mathcal{A}$-harmonic Green's function for the complement of a convex compact set $E$ when $n\leq p<\infty$. If $μ_E$ denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. We also show that this problem has a unique solution up to translation.
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Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
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.
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Absolute continuity of harmonic measure for domains with lower regular boundaries
We study absolute continuity of harmonic measure with respect to surface measure on domains $Ω$ that have large complements. We show that if $Γ\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $ω_Ω\ll \mathscr{H}^{d}$ on $Γ\cap \partialΩ$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $Γ\cap\partialΩ$ is a cone point if $Γ$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. Finally, we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.
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$σ$-finiteness of elliptic measures for quasilinear elliptic PDE in space
In this paper we study the Hausdorff dimension of a elliptic measure $μ_{f}$ in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $σ-$finite $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension $\leq n-1-δ$ for $p\geq n$ for some $δ$ which depends on various constants including $p$. The first result generalizes the authors previous work when the PDE is the $p-$Laplacian and the second result generalizes the well known theorem of Wolff when $p=2$ and $n=2$.
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On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains
In this paper, we study the set of absolute continuity of p-harmonic measure, $μ$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with $2 0=\mathcal{H}^{n-1}(K)$ where $μ$ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in $Ω$ with continuous boundary value zero on $\partialΩ$. We also show that there exists such a domain for which the same result holds when $p$ is fixed with $2-η 0$ provided that $n\geq 3$. This work is a generalization of a recent result of Azzam, Mourgoglou, and Tolsa when the measure $μ$ is harmonic measure at $x$, $ω=ω^{x}$, associated to the Laplace equation, i.e when $p=2$.
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Hausdorff dimension and $σ$ finiteness of $p-$harmonic measures in space when $p\geq n$
In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion Γof boundary of O. If p>n we show that this p harmonic measure is concentrated on a set of σ- finite H^{n-1} measure while if p=n the same conclusion holds provided Γis uniformly fat in the sense of n capacity.
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On the dimension of a certain measure in the plane
We study the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. Our work generalizes work of Lewis and coauthors when the measure is $p$ harmonic and also for $p=2$, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in a simply connected domain.
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