Elliptic Operators In Divergence Form Research Articles

An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

AbstractThe classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$, $u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$ and $s>n/2$. The inequality fails for $s=n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if $n\geqslant 3$ and $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$ is bounded, then $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $L^{p,q}$ is the Lorentz space refinement of $L^{p}$. This inequality fails for $n=2$, and we prove a sharp substitute result: there exists $c>0$ such that for all $\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$ with finite measure, $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$ This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

Parabolic equations with singular divergence‐free drift vector fields

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;\xi,\tau)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm $\left\Vert u\right\Vert _{L^{\infty}(\textrm{BMO}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $\left(L-\partial_{t}\right)v=0$ are established for initial data in $L^{2}$-space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form.

Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

On the homogenization of random stationary elliptic operators in divergence form

In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\nabla \cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probability measures to show that the law of $A_h$ may be obtained by using the standard homogenization results on each probability measure of the previous family. We finally provide a more explicit formula for $A_h$ in the case of coefficient fields which are a function of a stationary Gaussian field.

Regularity theory for solutions to second order elliptic operators with complex coefficients and the Lp Dirichlet problem

We establish a new theory of regularity for elliptic complex valued second order equations of the form L=divA(∇⋅), when the coefficients of the matrix A satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as Lp-dissipativity. Precisely, the regularity result is a reverse Hölder condition for Lp averages of solutions on interior balls, and serves as a replacement for the De Giorgi–Nash–Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for Lp-dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragičević introduced a condition they termed p-ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their p-ellipticity condition is exactly our strengthened version of Lp-dissipativity. The regularity results of the present paper are applied to solve Lp Dirichlet problems for L=divA(∇⋅)+B⋅∇ when A and B satisfy a Carleson measure condition, which previously was known only in the real valued case. We show solvability of the L2 Dirichlet problem, as well as solvability of the Lp Dirichlet boundary value problem for p in the range where A is p-elliptic.

Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_{k}(x):= \left\{\begin{array}{llll} \alpha_{k} \delta^{ij}(x) &x_{n}>0 \beta_{k} \delta^{ij}(x) &x_{n}<0. \end{array}\right. $$ In particular, we will show that the mean value sets associated to such an operator need not be convex as αk and βk converge to 1. This example then leads to an example of nonconvex mean value sets for smooth aij(x).

Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n|_{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse H\"older classes, that is, if for some $1<p<\infty$, one has $\omega_{L_0}\in RH_p(\sigma)$ then $\omega_{L}\in RH_p(\sigma)$. Equivalently, if the Dirichlet problem with data in $L^{p'}(\sigma)$ is solvable for $L_0$ then so it is for $L$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty(\sigma)$ then necessarily $\Omega$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable.

Long-time homogenization and asymptotic ballistic transport of classical waves

Consider an elliptic operator in divergence form with symmetric coefficients. If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization. When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associated operator are unclear. At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization. The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an ``eigendefect''), which we express with the help of a new family of extended correctors. We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times for the wave equation in terms of the spatial growth of these extended correctors. On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions). On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.

Representation formulae of solutions of second order elliptic inequalities

We prove various representation formulae of solutions of second order differential inequalities on RN. As an outcome we obtain positivity results of the solutions of −Lu≥0, where L is a general second order elliptic operator in divergence form. Results of this type are very useful for studying related Liouville theorems of second order semilinear inequalities.

Failure of $$L^2$$ L 2 boundedness of gradients of single layer potentials for measures with zero low density

Consider a totally irregular measure \(\mu \) in \({{\mathbb {R}}}^{n+1}\), that is, the upper density \(\limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) is positive \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\), and the lower density \(\liminf _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) vanishes \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\). We show that if \(T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu (y)\) is an operator whose kernel \(K(\cdot ,\cdot )\) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Holder continuous coefficients, then \(T_\mu \) is not bounded in \(L^2(\mu )\). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.

Eigenvalue estimates for a class of elliptic differential operators in divergence form

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl’s asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of Pólya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space.

Double phase anisotropic variational problems and combined effects of reaction and absorption terms

This paper deals with the existence of multiple solutions for the quasilinear equation −divA(x,∇u)+V(x)|u|α(x)−2u=f(x,u) in RN, which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like |ξ|q(x)−2ξ for small |ξ| and like |ξ|p(x)−2ξ for large |ξ|, where 1<α(⋅)≤p(⋅)<q(⋅)<N. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Rădulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p(⋅) and q(⋅) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Nonautonomous maximal Lp-regularity under fractional Sobolev regularity in time

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H\"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(\Omega)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{\alpha,p}$-regularity for $\alpha > \frac{1}{2}$ in time.

One-dimensional stochastic heat equation with discontinuous conductance

ABSTRACTWe introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.

Weighted Hardy Spaces Associated with Elliptic Operators. Part III: Characterisations of $$H_{L}^{p}({w})$$ H L p ( w ) and the Weighted Hardy Space Associated with the Riesz Transform

We consider Muckenhoupt weights w, and define weighted Hardy spaces \(H^p_{\mathcal {T}}(w)\), where \(\mathcal {T}\) denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second-order divergence form elliptic operator L. In the range \(0<p< 1\), we give a molecular characterisation of these spaces. Additionally, in the range \(p\in \mathcal {W}_w(p_-(L),p_+(L))\), we see that these spaces are isomorphic to the \(L^p(w)\) spaces. We also consider the Riesz transform \(\nabla L^{-\frac{1}{2}}\), associated with L, and show that the Hardy spaces \(H^p_{\nabla L^{-1/2},q}(w)\) and \(H^p_{\mathcal {S}_{ \mathrm {H} },q}(w)\) are isomorphic, in some range of \(p'\)s, and \(q\in \mathcal {W}_w(q_-(L),q_+(L))\).

Gradient estimates for double phase problems with irregular obstacles

An irregular obstacle problem with non-uniformly elliptic operator in divergence form of (p,q)-growth is studied. We find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence. Calderón–Zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data.

A local Faber–Krahn inequality and applications to Schrödinger equations

ABSTRACTWe prove a local Faber–Krahn inequality for solutions u to the Dirichlet problem for Δ+V on an arbitrary domain Ω in ℝn. Suppose a solution u assumes a global maximum at some point x0∈Ω and u(x0)>0. Let T(x0) be the smallest time at which a Brownian motion, started at x0, has exited the domain Ω with probability ≥1∕2. For nice (e.g., convex) domains, but we make no assumption on the geometry of the domain. Our main result is that there exists a ball B of radius such thatprovided that n≥3. In the case n = 2, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.

BMO Solvability and Absolute Continuity of Harmonic Measure

We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $\Omega$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial \Omega$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $\Omega$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $\Omega$ satisfies an interior Corkscrew condition, in the special case that $L$ is the Laplacian.

Boundary rectifiability and elliptic operators with W 1,1 coefficients

Abstract We consider second-order divergence form elliptic operators with W 1 , 1 {W^{1,1}} coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A ∞ {A_{\infty}} property of the elliptic measure associated to any such operator implies that Ω is a set of locally finite perimeter whose boundary, ∂ ⁡ Ω {\partial\Omega} , is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about Ω.