Elliptic Operators In Divergence Form Research Articles

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

AbstractIn the present paper we study perturbation theory for the $$L^p$$ L p Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $$L_0 = \text {div}(A_0\nabla )$$ L 0 = div ( A 0 ∇ ) and $$L_1 = \text {div}(A_1\nabla )$$ L 1 = div ( A 1 ∇ ) such that the $$L^p$$ L p Dirichlet problem for $$L_0$$ L 0 is solvable for some $$p>1$$ p > 1 ; we show that if $$A_0 - A_1$$ A 0 - A 1 satisfies certain Carleson condition, then the $$L^q$$ L q Dirichlet problem for $$L_1$$ L 1 is solvable for some $$q \ge p$$ q ≥ p . Moreover if the Carleson norm is small then we may take $$q=p$$ q = p . We use the approach first introduced in Fefferman–Kenig–Pipher ’91 on the unit ball, and build on Milakis–Pipher–Toro ’11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $$L^p$$ L p Dirichlet problem on a bounded Lipschitz domain for an operator $$L = \text {div}(A\nabla )$$ L = div ( A ∇ ) , where A satisfies a Carleson condition similar to the one assumed in Kenig–Pipher ’01 and Dindoš–Petermichl–Pipher ’07 but with unbounded antisymmetric part.

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form Lu = -{{textrm{div}}}(A cdot nabla u + b u ) - c cdot nabla u - du, in an open set Omega subset mathbb {R}^n, n ge 3, with possibly infinite Lebesgue measure. We assume that the n times n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c in L^{n,infty }_{text {loc}}({Omega }) and d in L_{text {loc}}^{frac{n}{2}, infty }(Omega ), or |b|^2, |c|^2, |d| in mathcal {K}_{text {loc}}(Omega ), where mathcal {K}_{text {loc}}(Omega ) stands for the local Stummel–Kato class. Let {mathcal {K}_{text {Dini}}}(Omega ) be a variant of mathcal {K}(Omega ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - {{textrm{div}}}g, where f and |g|^2 in {mathcal {K}_{text {Dini}}}(Omega ) if, for q in [n, infty ), any of the following assumptions holds: (i) |b|^2, |d| in {mathcal {K}_{text {Dini}}}(Omega ) and either c in L^{n,q}_{text {loc}}(Omega ) or |c|^2 in mathcal {K}_{text {loc}}(Omega ); (ii) {{textrm{div}}}b +d le 0 and either b+c in L^{n,q}_{text {loc}}(Omega ) or |b+c|^2 in mathcal {K}_{text {loc}}(Omega ); (iii) -{{textrm{div}}}c + d le 0 and |b+c|^2 in {mathcal {K}_{text {Dini}}}(Omega ). We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -{{textrm{div}}}c +d le 0, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|^2 in mathcal {K}'(Omega ), we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Omega , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

G-convergence of elliptic operators in non divergence form in R^n

The aim of this note is to prove a characterization of the G-limit of a sequence of elliptic operators in non-divergence form. As we consider any dimension, for this class of operators, it is not enough to deal with measurable and bounded coefficients so we need extra regularity assumptions on them.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/71/abstr.html

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace-Beltrami and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors.

Elliptic operators with discontinuous coefficients in meshfree GFDM

AbstractIn phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders of magnitude. These jump discontinuities represent interfaces between the phases, and they emerge naturally during the simulation; thus, their exact location is generally unknown a priori. Our goal is to simulate phase change processes with a meshfree generalized finite difference method in a monolithic model without distinguishing between the different phases. There, the material properties mentioned above appear as coefficients inside elliptic operators in divergence form and the jumps must be treated adequately by the numerical method. We present a numerical method for discretizing elliptic operators with discontinuous coefficients without the need for a domain decomposition or tracking of interfaces. Our method facilitates the construction of diagonally dominant diffusion operators that lead to M‐matrices for the discrete Poisson's equation, and thus, satisfy the discrete maximum principle. We demonstrate the applicability of the new method for the case of smooth diffusivity and discontinuous diffusivity. We show that the method is first‐order accurate for discontinuous diffusion problems and provides second‐order and fourth‐order convergence for continuous diffusion coefficients.

On approximation of the Dirichlet problem for divergence form operators by Robin problems

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.

Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.

Gradient estimates for the insulated conductivity problem with inclusions of the general m‐convex shapes

AbstractIn this paper, the insulated conductivity model with two touching or close‐to‐touching inclusions is considered in with . We establish the pointwise upper bounds on the gradient of the solution for the generalized m‐convex inclusions under these two cases with , which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non‐zero eigenvalue of an elliptic operator of divergence form on . Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.

Heat kernels and Hardy spaces on non-tangentially accessible domains with applications to global regularity of inhomogeneous Dirichlet problems

Let n ≥ 2 n\ge 2 and Ω \Omega be a bounded non-tangentially accessible domain (for short, NTA domain) of R n \mathbb {R}^n . Assume that L D L_D is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on L 2 ( Ω ) L^2(\Omega ) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} generated by L D L_D are Hölder continuous. Second, for any p ∈ ( 0 , 1 ] p\in (0,1] , the authors introduce the ‘geometrical’ Hardy space H r p ( Ω ) H^p_r(\Omega ) by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) to Ω \Omega and the authors show that, when p ∈ ( n n + δ 0 , 1 ] p\in (\frac {n}{n+\delta _0},1] , H r p ( Ω ) = H p ( Ω ) = H L D p ( Ω ) H^p_r(\Omega )=H^p(\Omega )=H^p_{L_D}(\Omega ) with equivalent quasi-norms, where H p ( Ω ) H^p(\Omega ) and H L D p ( Ω ) H^p_{L_D}(\Omega ) respectively denote the Hardy space on Ω \Omega and the Hardy space associated with L D L_D and where δ 0 ∈ ( 0 , 1 ] \delta _0\in (0,1] is the critical index of the Hölder continuity for the kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} . Third, as applications, the authors obtain the global gradient estimates in both L p ( Ω ) L^p(\Omega ) , with p ∈ ( 1 , p 0 ) p\in (1,p_0) , and H z p ( Ω ) H^p_z(\Omega ) , with p ∈ ( n n + 1 , 1 ] p\in (\frac {n}{n+1},1] , for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where p 0 ∈ ( 2 , ∞ ) p_0\in (2,\infty ) is a constant depending only on n n , Ω \Omega , and the coefficient matrix of L D L_D . Here, the ‘geometrical’ Hardy space H z p ( Ω ) H^p_z(\Omega ) is defined by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) supported in Ω ¯ \overline {\Omega } to Ω \Omega , where Ω ¯ \overline {\Omega } denotes the closure of Ω \Omega in R n \mathbb {R}^n . It is worth pointing out that the range p ∈ ( 1 , p 0 ) p\in (1,p_0) for the global gradient estimate in the scale of Lebesgue spaces L p ( Ω ) L^p(\Omega ) is sharp and the above results are established without any additional assumptions on both the coefficient matrix of L D L_D and the domain Ω \Omega .

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Optimal Poisson kernel regularity for elliptic operators with Hölder continuous coefficients in vanishing chord-arc domains

We show that if Ω is a vanishing chord-arc domain and L is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then log⁡kL∈VMO, where kL is the elliptic Poisson kernel for L in the domain Ω. This extends the previous work of Kenig and Toro in the case of the Laplacian.

Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations

We develop a new structure of the Green’s function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity $$\begin{aligned} w=\frac{1}{\varepsilon ^2}f_\varepsilon \left( \mathcal {G}_{K_H}w-\frac{\alpha }{2}|x|^2|\ln \varepsilon |\right) \ \ \text {in}\ \Omega \end{aligned}$$ for small $$ \varepsilon >0 $$ and considering a certain maximization problem for the vorticity, where $$ \mathcal {G}_{K_H} $$ is the inverse of an elliptic operator $$ \mathcal {L}_{K_H} $$ in divergence form, we get the existence of a family of concentrated helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under $$ L^p $$ perturbation when $$ p\ge 2. $$

Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called “h-version’’ Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

G-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Stochastic homogenization: a short proof of the annealed Calderón–Zygmund estimate

A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb {R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón–Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{{L}}^p(\mathbb {R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for p not close to 2. We seek an intrinsic criterion on the validity of the Calderón–Zygmund estimate in the whole range of \(p\in (1,\infty )\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón–Zygmund estimates we obtain are annealed as opposed to quenched, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{{L}}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of Avellaneda and Lin (Commun Pure Appl Math 40(6):803–847, 1987). More specifically, we provide an easier version of the proof given by us in Josien and Otto (J Funct Anal, 2022), albeit under stronger assumptions. Annealed Calderon–Zygmund estimates were first established in Duerinckx and Otto (Stoch Partial Differ Equ Anal Comput 8(3):625–692, 2020), and are a very convenient tool for error estimates in stochastic homogenization.

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for p_0 in [1,2) and q_0 in (2,infty ], we prove an estimate of the form where q_{0}^{*} is the Hölder conjugate of q_{0}/2, M is the underlying doubling space and {mathcal {S}} is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L^{p}(w).

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

Critical perturbations for second-order elliptic operators, I: Square function bounds for layer potentials

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.