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, the sharp range of $p$ is also found in the case $n=6$, 7 if $\ell=2$, and $n=2\ell+2$ if $\ell\ge 3$.
This paper studies the solvability of a class of Dirichlet problem associated with non-linear integro-differential operator. The main ingredient is the probabilistic construction of continuous supersolution via the identification of the continuity set of the exit time operators under Skorohod topology.
In the paper obtained equivalent system of Fredholm integral equations in the study of the Dirichlet problem for the generalized Manjeron equation with non-smooth coefficients in non-classical treatment (1), (4). When non-smooth conditions on the coefficients of the equation in a rectangular region for this problem posed are found correct solvability conditions in integral form based on the method of integral representations.
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint $BMO$ Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the $BMO$ Dirichlet problem implies $L^p$ solvability for all $p>p_0$.
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…
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 establish two commutator estimates for the Dirichlet-to-Neumann map associated with a second-order elliptic system in divergence form in Lipschitz domains. Our approach is based on Dahlberg's bilinear estimates.
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 introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
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 $2-\varepsilon<p<\frac{2(d-1)}{d-3}+\varepsilon$.
We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.
We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $Ω\subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincaré-Sobolev inequalities.
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…
In the paper, we establish commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains. The approach is based on Dahlberg's bilinear estimates, and the results may be regarded as an extension of [Dahlberg, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc., 97(1986), no.1, 41-48.] and [Shen, Commutator estimates for the Dirichlet-to-Neumann map in Lipschitz domains, Some topics in harmonic analysis and applications, 369C384, Adv. Lect. Math. (ALM)34, Int. Press, Somerville, MA, 2016.] to Stokes systems.
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 consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.
In this paper, we study a Dirichlet problem of a fractional Laplace equation in a bounded Lipschitz domain in $ \R, n \geq 2$. Our main result is that for the given data $F \in \dot H^s(\Om^c), 0 < s<1$, we find the function which satisfies that $\De^s u =0$ in $\Om$, $u|_{\Om^c} =F$ and $|u|_{\dot{H}^s(\R)} \leq c |F|_{\dot H^s(\Om^c)}$. Furthermore, we represent the solution with an integral operator.
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap betwe…
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…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a $\mathscr{C}^2$ bounded domain $Ω$ in $M$ and a function $ H = H (x, z) $ continuous in $\overlineΩ\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the strong Serrin condition $(n-1)\mathcal…
Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $Ω$ in $\mathbb{R}^d$ is solvable for some $1<p=p_0< \frac{2(d-1)}{d-2}$, then it is solvable for all $p$ satisfying $$ p_0<p< \frac{2(d-1)}{d-2} +\varepsilon. $$ The proof is based on a real-variable argument. It only requires that local solutions of $\mathcal{L}(u)=0$ satisfy a boundary Cacciopoli inequality.
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $Ω\subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $Ω$, with boundary data in Besov spaces. The main hypothesis under which our principal result is established is in the nature of best possible and requires that, at small scales, the mean oscillations of the unit normal to $\partialΩ$ and of the coefficients of the differential operator ${\mathcal L}(x,D_x)$ are not too large.
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…
In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, $ω\in A_\…
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain $Ω$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\mathbf{b}:Ω\rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b} \in L^{n}(Ω)^n$, we first establish existence and uniqueness of solutions in $L_α^{p}(Ω)$ for the Dirichlet and Neumann problems. Here $L_α^{p}(Ω)$ denotes the Sobolev space (or Bessel potential space) with the pair $(α,p…
It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$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 obtain…
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 …
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）
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 $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
In this paper we consider the problem of determining an unknown pair $λ$, $μ$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $λ$ and $μ$ from the Dirichlet to Neumann map.
We establish pointwise estimates for the Green function to the Dirichlet problem for parabolic equation with coefficients measurable in time variable. Using these estimate we obtain coercive estimates for this problem in anisotropic weighted Lebesgue spaces and prove the solvability theorems.
Let $Ω\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partialΩ$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of $Ω$, and hence that harmonic measure for $Ω$ is weak-$A_\infty$ with respect to surface measure on $\partialΩ$, provided that $Ω$ satisfies a certain weak version of a local John condition. Under the further assumption that $Ω$ satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa…
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 prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
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
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…
This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an $L^\infty$-tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in ${\mathbb R}^{n\times n}$ with zero matrix traces. We analyze, in $L^2$-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a Lipschitz interface in ${\mathbb R}^n$, $n\ge 2$ ($n=2,3$ for the nonlinear problems). The transversal interface inter…
