arXiv subjects

Martin Dindoš

Publications and source records attributed to Martin Dindoš.

At least 19 recordsLinked to original sources

arXiv2026

The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition

In this paper, we resolve the question of whether the Neumann problem for the parabolic PDE $-\partial_tu + \mathrm{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic with bounded and measurable coefficients that satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for any $1<p<\infty$ the Neumann problem is solvable under the assumption that both the Carleson norm of coefficients and the Lipschitz constant of the domain are sufficiently small (with dependence on $p$). The question of what happens in the "large Carleson norm/large Lipschitz constant" regime remains open, and even for elliptic PDEs this question has only been resolved in two dimensions. This paper complements results from our recent manuscript (by the same authors) in which the parabolic regularity problem has been fully resolved in both the small and large Carleson norm regime. Previously, the Dirichlet problem had been resolved under the same conditions by various authors.

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arXiv2026

The Regularity datum on time-varying graph domains and Dirichlet--Regularity duality

In this paper we address a basic question that has not been considered in the literature on the parabolic Regularity problem. Let $Ω$ be the region above the graph of a $\mathrm{Lip}(1,\tfrac12)$ function $ϕ$ on $\mathbb R^{n-1}\times\mathbb R$. The corresponding $\dot L^p_{1,1/2}$ Regularity boundary space on $\partialΩ$ is defined by pulling the datum back to the flat space where we have the norm $\|\nabla f\|_{L^p}+\|D^{1/2}_t f\|_{L^p}$. As there is more than one graph parametrisation of such a domain, this could petentially lead to different notions of $\dot L^p_{1,1/2}(\partialΩ)$. This issue arises only on time-varying domains, which is likely why the issue has not been noticed before. We show by example that the boundary spaces can indeed disagree. We then prove that an additional assumption $D^{1/2}_tϕ\in\mathrm{BMOpar}$ (called the Lewis--Murray condition), resolves the issue completely: the spaces defined through different graphs agree.\medskip This gives us a well-defined Regularity datum space on every Lewis--Murray graph domain. We then extend the duality between the parabolic Dirichlet and Regularity problems, proved on cylinders by the author and E.~Nyström, to such domains.

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arXiv2026

Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $Ω=\{(x',x_n):\,x_n>φ(x')\}\times\mathbb R$, where $φ:\mathbb R^{n-1}\to\mathbb R$ is a Lipschitz function. The result shows that under the assumptions that the $L^p$ parabolic Neumann problem for the equation $Lu=-\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ and also the $L^{p'}$ parabolic Dirichlet problem for the adjoint equation $L^*u=\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ are solvable, then also the $L^q$ parabolic Neumann problem for the equation $Lu=0$ in $Ω$ is solvable for all $1<q<p$. However the mentioned paper does not answer the question whether the same claim is also true for domains of the form $\mathcal O\times\mathbb R$, where $\mathcal O$ is a bounded Lipschitz domain (in spatial variables) since this case does not follow from our argument for the unbounded case. Indeed, the bounded Lipschitz cylinder case requires a significantly different approach which we present in this article and establish an analogous result when $\mathcal O$ is a bounded Lipschitz domain.

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arXiv2026

The $L^p$ regularity problem for parabolic operators with transversally independent coefficients

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients are independent of the spatial variable $x_{n+1}$ (which is transversal to the boundary). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. An analogous result for the Dirichlet problem has been considered earlier by Auscher, Egert and Nyström, however the Regularity problem represents an additional step up in difficulty. In the elliptic case, the analog of the question considered here was resolved for both Dirichlet and Regularity problems by Hofmann, Kenig, Mayboroda and Pipher. The main result of this paper complements a recent work of two of the authors with L. Li showing solvability of the parabolic Regularity problem for data in some $L^p$ spaces when the coefficients satisfy a natural Carleson condition (which is a parabolic analog of the so-called DKP-condition).

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arXiv2026

The $L^p$ regularity problem for parabolic operators

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

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arXiv2026

Localization and interpolation of parabolic $L^p$ Neumann problems

We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the extrapolation of solvability of the $L^p$ Neumann problem.

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arXiv2026

The Green Function for Elliptic Systems in the Upper-Half Space

Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the system $L$ in the upper-half space. Starting with a definition of the Green function which brings forth the minimal features which identify this object uniquely, we establish optimal nontangential maximal function estimates and regularity results up to the boundary for the said Green function. The main tools employed in the proof include the Agmon-Douglis-Nirenberg construction of a Poisson kernel for the system $L$, the Agmon-Douglis-Nirenberg a priori regularity estimates near the boundary, and the brand of Divergence Theorem from the book Geometric Harmonic Analysis Vol. I by the last three authors of this paper in which the boundary trace of the corresponding vector field is taken in nontangential pointwise sense.

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arXiv2025

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \mbox{div}(A_0\nabla)$ and $L_1 = \mbox{div}(A_1\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindoš-Petermichl-Pipher '07 but with unbounded antisymmetric part.

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arXiv2025

A relation between the Dirichlet and the Regularity problem for Parabolic equations

We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ Ω= \mathcal O \times \mathbb R $, where the base $ \mathcal O \subset \mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*)_{p'} $). We show that this holds if for at least of $q\in(1,\infty)$ the problem $ (R_L)_{q} $ is solvable. That is, we establish a duality/dichotomy result: Dirichlet solvability implies Regularity solvability in the dual $L^p$ range, or the Regularity problem is not solvable in any $L^p$. Results like these were only known in the elliptic settings (Kenig-Pipher (1993) and Shen (2006)) but are new for parabolic PDEs. Our result is one of the key components needed for the recent advancement of Dindoš, Li and Pipher in understanding solvability of the Regularity problem for operators whose coefficients satisfy certain natural Carleson condition (called also DKP-condition in the elliptic case).

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arXiv2025

On the regularity problem for parabolic operators and the role of half-time derivative

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $Ω=\mathcal O\times\mathbb R$ where $\mathcal O\subset\mathbb R^n$ is a is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$-Ahlfors regular. Let $u$ be a solution of such PDE in $Ω$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partialΩ)$ for some $p>1$. Furthermore, assume that for $u|_{\partialΩ}=f$ we have that $D^{1/2}_tf\in L^p(\partialΩ)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partialΩ)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.

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arXiv2022

Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition

In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the $L^q$ Dirichlet problem is solvable for some $1 < q < \infty$. Moreover, further studies completely resolved the range of $L^q$ solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small. We show that there exists $p_{reg}>1$ such that for all $1 1$ is the number such that the $L^q$ Dirichlet problem for the adjoint operator $\mathcal L^*$ is solvable for all $q>q_*$. Additionally when $n=2$, there exists $p_{neum}>1$ such that for all $1 1$ is the number such that the $L^q$ Dirichlet problem for the operator $\mathcal L_1=\mbox{div}(A_1\nabla\cdot)$ with matrix $A_1=A/\det{A}$ is solvable for all $q>q^*$.

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arXiv2022

Boundary value problems for elliptic operators satisfying Carleson condition

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with $L=\mbox{div}(A\nabla)$, we assume the matrix $A$ is elliptic and satisfies a natural Carleson condition either in the form that ($|\nabla A(X)|\lesssim \mbox{dist}(X,\partialΩ)^{-1}$ and $|\nabla A|(X)^2\mbox{dist}(X,\partialΩ)\,dX$) or $\mbox{dist}(X,\partialΩ)^{-1}\left(\mbox{osc}_{B(X,δ(X)/2)}A\right)^2\,dX$ is a Carleson measure. We present two types of results, the first is the so-called "small Carleson" case where, for a given $1<p<\infty$, we prove solvability of the three considered boundary value problems under assumption the Carleson norm of the coefficients and the Lipschitz constant of the considered domain is sufficiently small. The second type of results ("large Carleson") relaxes the constraints to any Lipschitz domain and to the assumption that the Carleson norm of the coefficients is merely bounded. In this case we have $L^p$ solvability for a range of $p$'s in a subinterval of $(1,\infty)$. At the end of the paper we give a brief overview of recent results on domains beyond Lipschitz such as uniform domains or chord-arc domains.

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arXiv2021

The $p$-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenisation problems

The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of $p$-ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity, rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when $p$-ellipticity is considered. In this paper, we define three new notions of $p$-ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems. These important roles are demonstrated by establishing extrapolation results for solvability of the $L^p$ Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization.

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arXiv2020

The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in an interval of the form $\big(2-\varepsilon,\frac{2(n-1)}{n-2}+\varepsilon\big)$ for some small $\varepsilon>0$). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity, instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lamé system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio $ν<0.396$. Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi - Nash - Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving $L^p$ solvability results for $p$ near $2$.

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arXiv2020

The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-\varepsilon < p<\frac{2(n-1)}{(n-3)}+\varepsilon$, for systems in dimension $n=2,3$ in the range $2-\varepsilon < p<\infty$. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in $L^p$) in the range $2-\varepsilon < p<2+\varepsilon$. Secondly, we look at perturbation type-results where we can deduce solvability of the $L^p$ Dirichlet problem for one operator from known $L^p$ Dirichlet solvability of a \lq\lq close" operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper arXiv:1708.02289; we establish solvability of the $L^p$ Dirichlet problem in the interval $2-\varepsilon < p<\frac{2(n-1)}{(n-2)}+\varepsilon$ under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lamé operator for isotropic inhomogeneous materials with Poisson ratio $ν<0.396$. In this specific case further improvements of the solvability range are possible, see the upcoming work with J. Li and J. Pipher.

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arXiv2020

Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients

In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as $p$-{\it ellipticity}. Specifically, let $Ω$ be a chord-arc domain in $\mathbb R^n$ and the operator $\mathcal L = \partial_{i}\left(A_{ij}(x)\partial_{j}\right) +B_{i}(x)\partial_{i} $ be elliptic, with $|B_i(x)| \le Kδ(x)^{-1}$ for a small $K$. Let $p_0 = \sup\{p>1: A \,\,\text{is}\,\, \text{$p$-elliptic}\}$. We establish that if the $L^q$ Dirichlet problem is solvable for $\mathcal L$ for some $1<q< \frac{p_0(n-1)}{(n-2)}$, then the $L^p$ Dirichlet problem is solvable for all $p$ in the range $[q, \frac{p_0(n-1)}{(n-2)})$. In particular, if the matrix $A$ is real, or $n=2$, the $L^p$ Dirichlet problem is solvable for $p$ in the range $[q, \infty)$.

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arXiv2018

Boundary value problems for second order elliptic operators with complex coefficients

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $|u|^{p/2-1}u \in W^{1,2}_{loc}$. These properties of solutions were used by Dindoš-Pipher to solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the Regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The Regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the Regularity results allow us to extend the previously established range of $L^p$ solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.

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arXiv2018

Perturbation theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div} A^0(x)\nabla+B^0(x)\cdot\nabla$ is a $p$-elliptic operator satisfying certain Carleson condition on $\nabla A$ and $B$ then the $L^p$ Dirichlet problem for the operator ${\mathcal L}_0$ is solvable in the upper half-space ${\mathbb R}^n_+$. In this paper we prove that the $L^p$ solvability is stable under small perturbations of ${\mathcal L}_0$. That is if ${\mathcal L}_1$ is another divergence form elliptic operator with complex coefficients and the coefficients of the operators ${\mathcal L}_0$ and ${\mathcal L}_1$ are sufficiently close in the sense of Carleson measures (considering the differences of coefficients), then the $L^p$ Dirichlet problem for the operator ${\mathcal L}_1$ is solvable for the same value of $p$. As a corollary we obtain a new result on $L^p$ solvability of the Dirichlet problem for operators of the form ${\mathcal L}=\mbox{div} A(x)\nabla+B(x)\cdot\nabla$ where the matrix $A$ satisfies weaker Carleson condition than in our earlier paper; in particular the coefficients of $A$ need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix $A$ over Whitney boxes. This result in the real case has been established by Dindoš, Petermichl and Pipher.

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