We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form: \[ \mathrm{div}(A(x)\nabla u)=0 \qquad\mathrm{in\}Ω, \] where $Ω$ is a bounded open subset of $\R^2$ and, for every $x\inΩ$, $A(x)$ is a matrix with bounded measurable coefficients. Such an estimate "interpolates" between the well-known estimate of Piccinini and Spagnolo in the isotropic case $A(x)=a(x)I$, where $a$ is a bounded measurable function, and our previous result in the unit determinant case $\det A(x)\equiv1$. Furthermore, we show that our estimate is sharp. Indeed, for every $τ\in[0,1]$ we construct coefficient matrices $A_τ$ such that $A_0$ is isotropic and $A_1$ has unit determinant, and such that our estimate for $A_τ$ reduces to an equality, for every $τ\in[0,1]$.
We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hoelder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Lad…
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the …
Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is discussed and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hoelder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hoelder regularity and the generalized thermoelastic plate equations.
This note presents a correction to the paper ``Hardy and BMO spaces associated to divergence form elliptic operators" by Steve Hofmann and Svitlana Mayboroda
We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implie…
This paper is devoted to establishing global $W^{2, p}$ estimate for strong solutions to the Dirichlet problem of uniformly elliptic equations in the non-divergence form where the domain is a Lipschitz polyhedra.
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
In 1957, De Giorgi [3] proved the Hölder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3].
We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has $C^{1,1}$ boundary. We also obtain pointwise bounds for the Green functions and its derivatives.
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma.
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence type elliptic equations set in halfspaces and with infinitely smooth data, may converge to their corresponding boundary layer tails as slow as one wish depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence type elliptic operators set in a bounded domain, and…
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetildeΩ \subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
We show that two classically known properties of positive supersolutions of uniformly elliptic PDEs, the boundary point principle (Hopf lemma) and global integrability, can be quantified with respect to each other. We obtain an extension to the boundary of the de Giorgi-Moser weak Harnack inequality, optimal with respect to the norms involved, for equations in divergence form.
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $Ω\subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
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 note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.
We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. From this, elliptic and parabolic regularity results are deduced by means of Sneiberg's isomorphism theorem.
In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special cases are analyzed. In conclusion, for completeness, a concrete example of an application is presented by finding the existence of three nontrivial weak solutions for an uniformly elliptic second-order problem on a bounded Euclidean domain.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet-Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish $L^2$ solvability of boundary value problems for a new class of the elliptic o…
We prove existence and regularity results for weak solutions of non linear elliptic systems with non variational structure satisfying $(p,q)$-growth conditions. In particular we are able to prove higher differentiability results under a dimension-free gap between $p$ and $q$.
In 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by Caffarelli, and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegene…
We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence region from non-existence. In the critical case $p=p^*$ we show that the existence of a singular solution depends on the rate at which the coefficients stabilize at zero and we provide some optimal conditions in this setting.
We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.
This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coe…
We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.
Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space of all continuous functions vanishing at the boundary. In particular, Lipschitz domains are allowed. This result was so far known under considerable stronger regularity assumptions. Also the Dirichlet problem is considered for such operators.
In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset Ω$) to the divergence-type equation $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F,$$ where $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $Ω\subset \mathbb{R}^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\mathbb{R}^N$, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer.
We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. Our results are new even for scalar equations and the proofs simplify the methods used previously in [12]
Consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calderón-Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some $L^p$ spaces. In this work we generalize the classical approach and develop a theory of Hardy and BMO spaces associated to $L$, which includes, in particular, molecular decomposition, maximal function characterization, duality of Hardy and BMO spaces, John-Nirenberg inequality, and allows to han…
In this paper, we show that $W^{1,p}$ $(1\leq p<\infty)$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^{1}$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower order coefficients are bounded, and interfacial boundaries are $C^{1,\text{Dini}}$. This extends a result of Li and Nirenberg (\textit{Comm. Pure Appl. Math.} \textbf{56} (2003), 892-925). Moreover, under a stronger assumption on the piecewise $L^{1}$-mean oscillation of the leading coefficients, we derive a global weak type-(1,1) estimate with respect to …
Paper dedicated to qualitative study of the solution of the Zaremba type problem in Lipschitz domain with respect to the elliptic equation in non-divergent form. Main result is Landis type Growth Lemma in spherical layer for Mixed Boundary Value Problem in the class of "admissible domain". Based on the Growth Lemma Phragmén-Lindelöf theorem is proved at junction point of Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers co…
We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $Ω\cap B_1$, $u_1=u_2=0$ in $B_1\setminusΩ$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $Ω$. When $Ω$ is Lipschitz, we show that the quotient $u_1/u_2$ is Hölder continuous up to the boundary in $B_{1/2}$.
Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.
In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these…
We study the existence of the Green function for an elliptic system in divergence form $-\nabla\cdot a\nabla$ in $\mathbb{R}^d$, with $d>2$. The tensor field $a=a(x)$ is only assumed to be bounded and $λ$-coercive. For almost every point $y \in \mathbb{R}^d$, the existence of a Green's function $G(a; \cdot, y)$ centered in $y$ has been proven in [J. Conlon, A. Giunti and F.Otto, "Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points $y \in \mathbb{R}^d$ for which $G(a; \cdot, y)$ does not exist has zero $p$-capacity, for an expon…
We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.
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 …
