A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
In this paper, we show that if the bounded solutions to the parabolic Dirichlet problem on a Lipshitz-$\left[1,\frac{1}{2}\right]$ domain obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to class $A^\infty$, which is equivalent to $L^p$ solvability for some $p<\infty$. This improves the existing literature which places additional assumptions on the parabolic uniform rectifiability or, equivalently, on the half-order time derivative of the function whose graph defines the boundary of the domain.
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.
We prove that the solvability of the regularity problem in $L^q(\partial Ω)$ is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same $L^q$, and if the perturbation is large, the regularity problem is solvable in $L^{r}$ for some other $r\in (1,\infty)$. We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular …
We establish $L^p$, $2\le p\le\infty$ solvability of the Dirichlet boundary value problem for a parabolic equation $u_t-\mbox{div}(A\nabla u)=0$ on time-varying domains with coefficient matrix $A=(a_{ij})$ that satisfy a small Carleson condition. The result is motivated by similar results for the elliptic equation $\mbox{div}(A\nabla u)=0$ that were established previously.
Let $Ω$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\partialΩ)$ and of the Neumann problem with $L^p(\partialΩ)$ data for the operator $L$ on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator $L$ to be rough obeying a certain Carleson condition with small norm. These results complete the results of [5] where $L^p(\partialΩ)$ Dirichlet problem was considered under the s…
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 bo…
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\,d…
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-\varepsil…
We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to study geometric and analytic properties of sets with boundaries whose co-dimension is higher than $1$. These operators are of the form $-\text{div} A\nabla$, where $A$ is a weighted elliptic matrix crafted to weigh the distance to the high co-dimension boundary in a way that allows for the nourishment of an elliptic theory. When this boundary is a $d-$Alhfors-D…
We study the homogeneous elliptic systems of order $2\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in $L^p$. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the $L^p$ Dirichlet problem for $n\ge 4$ and $2-\e< p<\frac{2(n-1)}{n-3} +\e$. The range of $p$ is known to be sharp if $\ell\ge 2$ and $4\le n\le 2\ell +1$. For the polyharmonic equation, th…
We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic functions under the assumption that the variable exponent $p$ is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for $p(\cdot)$-harmonic functions in NTA domains in $\mathbb{R}^n$ and provide lower- and upper- growth estimates and a doubling property for a $p(\cdot)$-harmonic …
We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on bounded solutions in the elliptic case, improving an earlier result of Dindos, Kenig and Pipher for solutions with data in BMO. The extension to the parabolic setting requires an approach to the key estimate that primarily exploits the maximum principle. For various classes of parabolic operators, this criterion makes it easier to establish the solvability of…
We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potent…
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…
This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^\prime}$ Dirichlet problem. For higher order elliptic equations and systems, we show that if $p>2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.
We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_μ(\mathbb{R}^n)$ for some $p<\infty$. The coefficient matrix $A$ is only assumed to be measurable, real-valued and $t$-independent with a degenerate bound and ellipticity controlled by an $A_2$-weight $μ$. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure …
We study the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is sufficient for the solvability of the Dirichlet problem with boundary data in $L^p_N(\partialΩ,\rn{d})$. Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability ofthe $L^p$ Dirichlet problem for $d\geq 4$ and $2-\varepsilon<p<\frac{2(d-1)}{d-3}+\varepsilon$.
The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)$
We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet bou…
In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal Carleson measure estimate for bounded solutions of such operators implies a regularity result for the associated elliptic measure.
In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\rm div}b$ in order to obtain solvability. O…
We establish Dahlberg's perturbation theorem for non-divergence form operators L = A\nabla^2. If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p in (1,\infty) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the L^p Dirichlet problem for the operator L_1 is solvable for the same p. This is an improvement of the A_{\infty} version of this result proved by Rios in "The L^p Diriclet problem and nondivergence harmonic measure" (Trans. AMS 355, 2 (2003)). As a c…
We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-ε<p<2$, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.
We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson…
We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $Ω$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $Ω$. As a result, we prove that for any given bounded Lipschitz domain $Ω$ in $\rn{d}$ and $1<q<\infty$, the solvability of the $L^{q}$ Dirichlet problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{1,q}(\partialΩ)$ is equivalent to that of the $L^p$ regularity problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{2,p}(\partialΩ)$, where $\frac{1}{p} +\frac{1}{q}=1$. This duality relation, together with…
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, …
Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph
Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains
Université Paris-Saclay（巴黎萨克雷大学）；The University of Edinburgh（爱丁堡大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14, 15], which work for $C^{1,α}$ domains with $α> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.
Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the solvability of the $L^p$ Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of $p$. In the case of convex domains, the estimates allow us to show that the $L^p$ Dirichlet problem is uniquely solvable for any $2-\e<p<\infty$ and $n\ge 4$.
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere no…
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
This article considers two weight estimates for the single layer potential --- corresponding to the Laplace operator in $\mathbf{R}^{N+1}$ --- on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we …
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 c…
We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class…
We consider relativistic plasma particles subjected to an external gravitation force in a $3$D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov-Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. We construct weak solutions to these initial-boundary v…
In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-Δ)^s u = αu^+ -βu^{-} + f(u) + h \; \text{in}\;Ω\quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n> 2s$, $0<s<1$, $(α, β) \in \mathbb{R}^2$, $f: \mathbb{R}\to \mathbb{R}$ is a bounded and continuous function and $h\in L^2(Ω)$. We prove the existence results in two cases: First, the nonresonance case, where $(α,β)$ is not an element of the Fučik spectrum. Second, the resonance case, w…
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $ω_L\in A_{\infty}(σ)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.
Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.
We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal dir…
Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable non-smooth (or limited-smoothness) coefficient on Lipschitz domains are formulated. The PDE right hand sides belong to the Sobolev (Bessel-potential) space $H^{s-2}(Ω)$ or $\widetilde H^{s-2}(Ω)$, $1/2<s<3/2$, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, so…
In the present paper we study the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains beyond the Lipschitz category. In our approach, the development of the theory of tent spaces on these domains is essential.
We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $Ω$ in $R^n$. For $n\ge 4$ and $2-ε<p<2(n-1)/(n-3)+ε$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partialΩ)$. In the case of the polyharmonic equation $Δ^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$.
