arXiv subjects

Mihalis Mourgoglou

Publications and source records attributed to Mihalis Mourgoglou.

At least 19 recordsLinked to original sources

arXiv2026

Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

Let $Ω\subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $Ω$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1 p$ in $Ω$, $\partial Ω$ supports a weak $p$-Poincaré inequality, and $Ω$ has very big pieces of superdomains for which the Neumann problem for $\mathcal L$ is solvable uniformly in $L^q$, then the Neumann problem for $\mathcal L$ is solvable in $L^p$ in $Ω$.

math.AP↗

arXiv2026

The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result

On a bounded domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_{2,p}$ of functions $u$ whose Kenig-Pipher modified non-tangential maximal operator $\mathcal N_2(u)$ lies in $L^p(\partialΩ)$, $p\in(1,\infty)$. We find that \[ ({\bf N}_{2,p})^*={\bf C}_{2,p'}\oplus L^{p'}(\partialΩ),\qquad\text{and that}\qquad L^{p'}(\partialΩ)=\partial^{\operatorname{weak}-*}{\bf C}_{2,p'}\,/\,{\bf C}_{2,p'}, \] where ${\bf C}_{2,p'}$ is a certain $L^{p'}$-Carleson space and $p'$ is the Hölder conjugate of $p$. This answers a question considered by Hytönen and Rosén. Inspired by this result and the recently understood characterizations of the $L^p$-solvability of the Dirichlet problem in terms of the Poisson problem by Mourgoglou, Poggi, and Tolsa, we show a novel approximation result: for an arbitrary elliptic operator $L=-\operatorname{div} A\nabla$ with a not necessarily symmetric matrix $A$ of real bounded measurable coefficients, the solution space to the Dirichlet problem with data in $L^p(\partialΩ)$ \[ \left\{\begin{aligned}-\operatorname{div} A\nabla u&=0,\quad&\text{in }&Ω,\\u&=g,\quad&\text{on }&\partialΩ,\end{aligned}\right. \] lies on the weak-$*$ boundary in ${\bf N}_{2,p}$ of the solution space to the Poisson problem \[ \left\{\begin{aligned}-\operatorname{div} A\nabla w&=-\operatorname{div} F,\qquad&\text{in }&Ω,\\ w&=0,\qquad&\text{on }&\partialΩ,\end{aligned}\right. \] with $F\in{\bf C}_{2,p}$, provided that the Dirichlet problem for $L$ with data in $L^p(\partialΩ)$ is solvable in $Ω$. This approximation result is sharp and new even for the Laplacian and on the unit ball.

math.AP↗

arXiv2025

Quantitative differentiability on uniformly rectifiable sets

We prove $L^p$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-type theorem holds in this context: the $L^p$ norm of the gradient of a Sobolev function $f: E \to \mathbb{R}$ is comparable to the $L^p$ norm of a new square function measuring both the affine deviation of $f$ and how flat the subset $E$ is. A corollary dealing with extensions and traces of Sobolev functions may be found in a companion article.

math.CA↗

arXiv2025

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Hajlasz-Sobolev space on the boundary from the space of functions on the domain with $L^p$ norm involving the non-tangential maximal function of the gradient and the conical square function of the Hessian. This fundametally uses the Dorronsoro theorem for UR sets proven in a companion paper.

math.CA↗

arXiv2025

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $Tb$ theorem, quantitative rectifiability, and free boundary problems

For $n \geq 2$, we consider the operator $L_A = -\mathrm{div }(A(\cdot)\nabla)$, where $A$ is a uniformly elliptic $(n+1)\times(n+1)$ matrix with variable coefficients, a Radon measure $μ$ on $\mathbb{R}^{n+1}$, and the associated gradient of the single layer potential operator $T_μ$. Under a Dini-type assumption on the mean oscillation of the matrix $A$, we establish the following results: 1) A rectifiability criterion for $μ$ in terms of $T_μ$. Under quantitative geometric and analytic assumptions within a ball $B$ -- including an upper $n$-growth condition on $μ$ in $B$, a thin boundary condition, a scale-invariant decay condition expressed via a weighted sum of densities over dyadic dilations of $B$, and $L^2$ boundedness of the gradient of $T_μ$ -- we show the following: if the support of $μ$ lies very close to an $n$-plane in $B$, and $T_μ1$ is nearly constant on $B$ in the $L^2$ sense, then there exists a uniformly $n$-rectifiable set $Γ$ such that $μ(B \cap Γ) \gtrsim μ(B)$. 2) A $Tb$ theorem for suppressed $T_μ$, which extends a well-known theorem of Nazarov, Treil, and Volberg, and holds also for a broader class of singular integral operators. These results make it possible to prove both qualitative and quantitative one- and two-phase free boundary problems for elliptic measure, formulated in terms of (uniform) rectifiability, in bounded Wiener-regular domains.

math.AP↗

arXiv2024

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem

We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\operatorname{div} A^T \nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$. This result for Dahlberg-Kenig-Pipher operators is new even if $Ω$ is the unit ball, despite the fact that the $L^{p'}$-solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001. Further novel applications include i) new local estimates for the Green's function and its gradient in rough domains, ii) a local $T1$-type theorem for the $L^{p}$-solvability of the ``Poisson-Regularity problem'', itself equivalent to the $L^{p'}$-solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and iii) new $L^p$ estimates for the eigenfunctions (and their gradients) of symmetric operators $L$ on bounded rough domains.

math.AP↗

arXiv2024

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with $L^\infty$ coefficients

Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 1$, be an open set with $s$-Ahlfors regular boundary $\partial Ω$, for some $s \in(0,n]$, such that either $s=n$ and $Ω$ is a corkscrew domain with the pointwise John condition, or $s<n$ and $Ω= \mathbb{R}^{n+1} \setminus E$, for some $s$-Ahlfors regular set $E \subset \mathbb{R}^{n+1}$. In this paper we provide a unifying method to construct Varopoulos' type extensions of $BMO$ and $L^p$ boundary functions. In particular, we show that a) if $ f \in BMO(\partial Ω)$, there exists $ F\in C^\infty(Ω)$ such that $dist(x, Ω^c)|\nabla F(x)|$ is uniformly bounded in $Ω$ and the Carleson functional of $dist(x,Ω^c)^{s-n}|\nabla F(x)|$ as well the sharp non-tangential maximal function of $ F$ are uniformly bounded on $\partial Ω$ with norms controlled by the $BMO$-norm of $ f$, and $ F \to f$ in a certain non-tangential sense $\mathcal H^s|_{\partial Ω}$-almost everywhere; b) if $\bar f \in L^p(\partial Ω)$, $1 <p \leq \infty$, there exists $\bar F \in C^\infty(Ω)$ such that the non-tangential maximal functions of $\bar F$ and $dist(\cdot, Ω^c)|\nabla \bar F|$ as well as the Carleson functional of $dist(\cdot,Ω^c)^{s-n}|\nabla \bar F|$ are in $L^p(\partial Ω)$ with norms controlled by the $L^p$-norm of $\bar f$, and $\bar F \to \bar f$ in some non-tangential sense $\mathcal H^s|_{\partial Ω}$-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support, then both $F$ and $\bar F$ can be constructed so that they are also Lipschitz on $\barΩ$ and converge to the boundary data continuously. The latter results hold without the additional assumption of the pointwise John condition. Finally, we give some applications of the constructed extensions in the connection between Poisson problems and BVPs.

math.AP↗

arXiv2023

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Let $Ω\subset \mathbb R^{n+1}$, $n\geq2$, be an open set satisfying the corkscrew condition with $n$-Ahlfors regular boundary $\partialΩ$, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space $M^{1,1}(\partialΩ)$ and the weak-$\mathcal A_\infty$ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in $M^{1,1}(\partialΩ)$ is equivalent to the solvability of the regularity problem in $M^{1,p}(\partialΩ)$ for some $p>1$. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that $\partialΩ$ supports a weak $(1,1)$-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space $M^{1,1}(\partialΩ)$ is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

math.AP↗

arXiv2023

Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\nabla u+bu)-c\nabla u-du$, in an open set $Ω\subset \mathbb{R}^n$, $n\geq 3$, with possibly infinite Lebesgue measure. We assume that the $n\times n$ matrix $A$ is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either $b,c\in L^{n,\infty}_{loc}(Ω)$ and $d\in L_{loc}^{\frac{n}{2},\infty}(Ω)$, or $|b|^2,|c|^2,|d|\in \mathcal{K}_{loc}(Ω)$, where $\mathcal{K}_{loc}(Ω)$ stands for the local Stummel-Kato class. Let $\mathcal{K}_{Dini}(Ω)$ be a variant of $\mathcal{K}(Ω)$ satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of $Lu=f-$div$g$, where $|f|$ and $|g|^2\in \mathcal{K}_{Dini}(Ω)$ if, for $q\in [n, \infty)$, any of the following assumptions holds: a) $|b|^2,|d|\in \mathcal{K}_{Dini}(Ω)$ and either $c\in L^{n,q}_{loc}(Ω)$ or $|c|^2\in \mathcal{K}_{loc}(Ω)$; b) div$b +d \leq 0$ and either $b+c\in L^{n,q}_{loc}(Ω)$ or $|b+c|^2\in \mathcal{K}_{loc}(Ω)$; c) $-$div$c+d \leq 0$ and $|b+c|^2\in \mathcal{K}_{Dini}(Ω)$. We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming $-$div$c+d\leq 0$, we construct the Green's function associated with $L$ satisfying quantitative estimates. Under the additional hypothesis $|b+c|^2\in \mathcal{K}'(Ω)$, we show that it satisfies global pointwise bounds and also construct the Green's function associated with the formal adjoint operator of $L$. An important feature of our results is that all the estimates are scale invariant and independent of $Ω$, while we do not assume smallness of the norms of the coefficients or coercivity of the bilinear form.

math.AP↗

arXiv2023

The regularity problem for the Laplace equation in rough domains

Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection between the solvability of $(D_{p'})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p'}(\partial Ω)$, and $(R_{p})$ (resp. $(\tilde R_{p})$), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space $W^{1,p}(\partial Ω)$ (resp. $\tilde W^{1,p}(\partial Ω)$, the usual Sobolev space in terms of the tangential derivative), where $p \in (1,2+\varepsilon)$ and $1/p+1/p'=1$. Our main result shows that $(D_{p'})$ is solvable if and only if so is $(R_{p})$. Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that $(D_{p'}) \Rightarrow (\tilde R_{p})$. In particular, we deduce that in bounded chord-arc domains (resp. two-sided chord-arc domains) there exists $p_0 \in (1,2+\varepsilon)$ so that $(R_{p_0})$ (resp. $(\tilde R_{p_0})$) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with $n$-Ahlfors-David regular boundaries the single layer potential operator is invertible from $L^p(\partial Ω)$ to the inhomogeneous Sobolev space $ W^{1,p}(\partial Ω)$. Finally, we provide a counterexample of a chord-arc domain $Ω_0 \subset \mathbb{R}^{n+1}$, $n \geq 3$, so that $(\tilde R_p)$ is not solvable for any $p \in [1, \infty)$.

math.AP↗

arXiv2022

On the density problem in the parabolic space

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $\mathbb{R}^{n+1}$ equipped with parabolic dilations. In particular we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of $1$-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant density condition for a $1$-codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma.

math.MG↗

arXiv2022

$L^2$-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix $A$ with real, merely bounded, and possibly non-symmetric coefficients. If $$ω_A(r)=\sup_{x\in \mathbb{R}^{n+1}} \frac{1}{|B(x,r)|}\int_{B(x,r)}\Big|A(z)-\frac{1}{|B(x,r)|}\int_{B(x,r)}A\Big|\,dz,$$ then, under suitable Dini-type assumptions on $ω_A$, we prove the following: if $μ$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, and $T_μf(x)=\int \nabla_xΓ_A (x,y)f(y)\, dμ(y)$ denotes the gradient of the single layer potential associated with $L_A$, then $$ 1+ \|T_μ\|_{L^2(μ)\to L^2(μ)}\approx 1+ \|\mathcal R_μ\|_{L^2(μ)\to L^2(μ)},$$ where $\mathcal R_μ$ indicates the $n$-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $\mathcal R_μ$, which were recently extended to $T_μ$ associated with $L_A$ with Hölder continuous coefficients. In particular, we show the following: 1) If $μ$ is an $n$-Ahlfors-David-regular measure on $\mathbb{R}^{n+1}$ with compact support, then $T_μ$ is bounded on $L^2(μ)$ if and only if $μ$ is uniformly $n$-rectifiable. 2) Let $E\subset \mathbb{R}^{n+1}$ be compact and $\mathcal H^n(E)<\infty$. If $T_{\mathcal H^n|_E}$ is bounded on $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. 3) If $μ\not\equiv 0$ satisfies $\limsup_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ {is positive and finite} for $μ$-a.e. $x\in \mathbb{R}^{n+1}$ and $\liminf_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ vanishes for $μ$-a.e. $x\in \mathbb{R}^{n+1}$, then $T_μ$ is not bounded on $L^2(μ)$. 4) If $μ$ is a compactly supported Radon measure satisfying a certain set of local conditions at the level of a ball $B$ with small radius, then a significant portion of $μ|_B$ can be covered by a UR set.

math.AP↗

arXiv2021

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: 1) Let $Ω_1$ and $Ω_2$ be disjoint domains in $\mathbb{R}^{n+1}$, $n \geq 2$, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set $E$ of mutual absolute continuity of the associated caloric measures $ω_i$ with poles at $\bar{p}_i=(p_i,t_i)\inΩ_i$, $i=1,2$. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of $ω_1|_E$ is $n+1$ and the tangent measures of $ω_1$ at $ω_1$-a.e. point of $E$ are equal to a constant multiple of the parabolic $(n+1)$-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis. 2) If, additionally, $ω_1$ and $ω_2$ are doubling, $\log \frac{dω_2|_E}{dω_1|_E} \in VMO(ω_1|_E)$, and $E$ is relatively open in the support of $ω_1$, then their tangent measures at {\it every} point of $E$ are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that in complementary $δ$-Reifenberg flat domains, if $δ$ is small enough and $\log \frac{dω_2}{dω_1} \in VMO(ω_1)$, then $Ω_1 \cap \{t<t_2\}$ is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian. 3) We establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure.

math.AP↗

arXiv2021

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

Let $Ω\subset \mathbb{R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal{T}_n$ to be the set of points $ξ\in \partial Ω$ so that there exists an approximate tangent $n$-plane for $\partialΩ$ at $ξ$ and $\partialΩ$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $ξ$. We first show that $\mathcal{T}_n$ can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting $\partial^\star Ω$ be a subset of $\mathcal{T}_n$ where $Ω$ satisfies an appropriate thickness condition, we prove that $\partial^\star Ω$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $Ω$. As a corollary we obtain that if $Ω$ has locally finite perimeter, $\partialΩ$ is weakly lower Ahlfors-David $n$-regular, and the measure-theoretic boundary coincides with the topological boundary of $Ω$ up to a set of $\mathcal{H}^n$-measure zero, then $\partial Ω$ can be covered, up to a set of $\mathcal{H}^n$-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in $Ω$. This implies that in such domains, $\mathcal{H}^n|_{\partialΩ}$ is absolutely continuous with respect to harmonic measure.

math.CA↗

arXiv2020

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ Ω\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $Ω$, with data in $L^p(\partialΩ)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.

math.CA↗

arXiv2020

A two-phase free boundary problem for harmonic measure and uniform rectifiability

We assume that $Ω_1, Ω_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure of $Ω_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $Ω_2$ in $B$, then there exists a uniformly $n$-rectifiable set $Σ$ so that the harmonic measure of $Σ\cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the harmonic measures do not need to satisfy any doubling condition. In the particular case that $Ω_1$ and $Ω_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective harmonic harmonic measures of $Ω_1$ and $Ω_2$.

math.AP↗

arXiv2019

A new approach to non-homogeneous local $Tb$ theorems

We develop a new general method to prove various non-doubling local Tb theorems. The method combines the non-homogeneous good lambda method of Tolsa, the big pieces Tb theorem of Nazarov-Treil-Volberg and a new change of measure argument based on stopping time techniques. We also improve known results and discuss some further applications.

math.CA↗

arXiv2018

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part II

Let $Ω\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $Ω$ satisfies the so-called weak-$A_\infty$ condition, then $Ω$ satisfies a suitable connectivity condition, namely the weak local John condition. Together with other previous results by Hofmann and Martell, this implies that the weak-$A_\infty$ condition for harmonic measure holds if and only if $\partialΩ$ is uniformly $n$-rectifiable and the weak local John condition is satisfied. This yields the first geometric characterization of the weak-$A_\infty$ condition for harmonic measure, which is important because of its connection with the Dirichlet problem for the Laplace equation.

math.AP↗