Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ symmetric part and small $\mathrm{BMO}$ anti-symmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $C^1$ domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz--)Morrey spaces, (Musielak--)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-s…
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 $p>p_0$.
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_\…
The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation ${\rm div}(A\nabla u)=0$ in the upper half-space $(x,t)\in\mathbb{R}^{n+1}_+$ is uniquely solvable when $n\ge2$ and the boundary data is in $L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$. This result is equivalent to saying that the elliptic…
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.
In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $Ω\subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$, the space of functions of Bounded Mean Oscillation of John & Nirenberg. A version of Taylor's theorem is first shown to be valid provided the integrand $W$ has polynomial growth. This result is then used to demonstrate that, for the Dirichlet, Neumann, and mixed problems, every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the se…
In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian $Δ_{N_+}$ on the upper half space $\mathbb{R}^n_+$ and the reflection Neumann Laplacian $Δ_N$ on $\mathbb{R}^n$ with respect to the weights associated to $Δ_{N_+}$ and $Δ_{N}$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to $Δ_{N}$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contribut…
In this paper, we investigate the boundary behavior of solutions of divergence-form operators with an elliptic symmetric part and a $BMO$ anti-symmetric part. Our results will hold in non-tangentially accessible (NTA) domains; these general domains were introduced by Jerison and Kenig and include the class of Lipschitz domains. We establish the Hölder continuity of the solutions at the boundary, existence of elliptic measures $ω_L$ associated to such operators, and the well-posedness of the continuous Dirichlet problem as well as the $L^p(dω)$ Dirichlet problem in NTA domains. The equivalence …
The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients $a^{ij}$ are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that $a^{ij}$ have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We …
We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space $BMO(\mathbb{R}^n)$, $n\ge2$. We prove the Gaussian estimates for the kernel of $e^{-tL}$, as well as that of $\partial_t^le^{-tL}$, for any $l\in\mathbb{N}$. We show that the square root of $L$ satisfies the $L^p$ estimates $\left\Vert{L^{1/2}f}\right\Vert_{L^p}\lesssim\left\Vert{\nabla f}\right\Vert_{L^p}$ for $1<p<\infty$, and $\left\Vert{\nabla f}\right\Vert_{L^p}…
We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients a_k^{ij} belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L_0 is regular in L^p(E,ds) for some p, 1<p<\infty, that is, |Nu|_{L^p}< C |g|_{L^p} for all continuous boundary data g. Here ds is the surface measure on E and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient cond…
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 study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
Weakly harmonic maps from a domain $Ω$ (the upper half-space $\Rd$ or a bounded $C^{1,α}$ domain, $α\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes $L^{\infty}(\partialΩ)$ or $BMO(\partialΩ)$, we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$. Moreover, we show that boundary data can be chosen large in the underlying topologies if $Ω$ is smooth and bounded by perturbing strictly stable smooth harmonic…
Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L}=-Δ+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class ${RH}_q$ for some $q\geq (n+1)/2$. Let ${CMO}_{\mathcal{L}}(\mathbb{R}^n)$ denote the function space of vanishing mean oscillation associated to $\mathcal{L}$. In this article we will show that a function $f$ of ${ CMO}_{\mathcal{L}}(\mathbb{R}^n) $ is the trace of the solution to $\mathbb{L}u=-u_{tt}+\mathcal{L} u=0$, $u(x,0)=f(x)$, if and only if, $u$ satisfies a Carleson condition $$ \sup_{B: \ { balls}}\mathcal{C}_{u…
Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $Δ_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of Perron's method and a priori estimates to give general sufficient conditions on the right hand side $f$ that would ensure existence of viscosity solutions to the Dirichlet problem. Examples show that these sufficient conditions may not be relaxed. We also identify a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any dom…
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 this paper we investigate elliptic partial differential equations on Lipschitz domains in the plane whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. We show that for Dirichlet boundary data in L^p for p large enough, solutions exist and are controlled by the L^p-norm of the boundary data. Similarly, for Neumann boundary data in L^q, or for Dirichlet boundary data whose tangential derivative is in L^q (regularity boundary data), for q small enough, we show that solutions exist and are controlled by the L^q-norm of the…
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces…
We show that the fractional wave operator, which is usually studied in the context of hypersingular integrals but had not yet appeared in mathematical physics, can be constructed as the Dirichlet-to-Neumann map associated with the Klein-Gordon equation in anti-de Sitter spacetimes. Several generalizations of this relation will be discussed too.
We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions. Thus we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We require a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary conditions. Our proof proceeds by defining a variant of the space $BMO(\partial Ω)$ that is adapted to the boundary conditions and showing tha…
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 investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity $C^{1, α}$, $0 < α\le 1$. First, we show the existence and uniqueness of strong solutions up to collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard $H^2$ estimate for smoother domains. Then, we study the asymptotic behaviour of one $C^{1, α}$ body falling over a flat surface. We show t…
We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $Ω$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum princi…
In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-Δ)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the …
An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the $p$-Laplace equation and system, with right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $\mathrm{BMO}$ and $\mathrm{VMO}$ spaces…
In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ Ω, \mbox{ and } \ u = g \ \text{on} \ \partial Ω, \end{align*} that related to Riesz potentials. Here, in our setting, $Ω\subset \mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded me…
A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón-Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical $(-Δ_p)^1$-Laplace …
The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ Δu+g(u)= μϕ_1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on $\partial D$} \,, \] depending on the real parameter $μ$, for a class of convex $g(u)$, and $\int _D e(x) ϕ_1(x)\, dx=0$ (where $ϕ_1(x)>0$ is the principal eigenfunction of the Laplacian on $D$, and $D \subset R^n$ is a smooth domain). By considering generalized harmonics, we give a similar result for the problem \[ Δu+g(u)= μf(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…
In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize …
In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.
The anti-maximum principle for the homogeneous Dirichlet problem to $-Δ_p u = λ|u|^{p-2}u + f(x)$ with positive $f \in L^\infty(Ω)$ states the existence of a critical value $λ_f > λ_1$ such that any solution of this problem with $λ\in (λ_1, λ_f)$ is strictly negative. In this paper, we give a variational upper bound for $λ_f$ and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem in $(λ_1,λ_2)$.
A recursive method for computing singular solutions in corners with homogeneous Dirichlet-Robin boundary condition with power-law coefficient variation
We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal eigenfunctions in viscosity sense, maximum principles, continuity property of the principal eigenvalues with respect to domains etc. We also prove an anti-maximum principle and study existence result for some nonlinear problem via Rabinowitz bifurcation-type results.
In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metric g^ on X, where C has a non-trivial boundary, in the sense that g=x^{-2}g^, with x a boundary defining function. The boundary is conformally time-like for these spaces, unlike asymptotically de Sitter spaces, which …
In this article we consider direct and inverse problems for $α$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as introduced in \cite{I86}. We consider the Dirichlet problem for these operators on the "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators …
