Elliptic Operators In Divergence Form Research Articles

Fractional stochastic heat equation with piecewise constant coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

Generic simplicity of spectral edges and applications to homogenization

We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.

Heat Kernels and Besov Spaces Associated with Second Order Divergence Form Elliptic Operators

Let $${\mathcal {L}}=-\mathrm {div} (A\nabla )$$ be a uniformly elliptic operator in $${\mathbb {R}}^n$$ with real, symmetric, measurable coefficients. We study the identity of two families of Besov spaces $$B_{p,q}^{s,{\mathcal {L}}}$$ and $$B_{p,q}^{s}$$, where the former one is defined using the heat semigroup of $${\mathcal {L}}$$, while the latter one is defined in a classical way, using the metric structure of $${\mathbb {R}}^n$$. A sharp range of parameters p, q, s ensuring the identity $$B_{p,q}^{s,{\mathcal {L}}}=B_{p,q}^{s}$$ is given by a Hardy–Littlewood–Sobolev–Kato diagram.

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Multilevel approximation of Gaussian random fields: Fast simulation

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

We investigate a 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. We introduce a notion of weak solution of this equation and prove its equivalence to the already known notion of mild solution.

Tangent measures of elliptic measure and applications

Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $ \Omega \subset \mathbb{R}^{n+1}$: 1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for {\it any} domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$. 2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. 3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if $\omega$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.

Inequalities for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and provide a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.

Riesz transform and fractional integral operators generated by nondegenerate elliptic differential operators

‎‎‎The Morrey boundedness is proved for the Riesz transform and the inverse operator of the nondegenerate elliptic differential operator of divergence form generated by a vector-function in $(L^\infty)^{n^2}$‎, ‎and for the inverse operator of the Schrodinger operators whose nonnegative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture did not use any integral expression of the operators‎. ‎One of the important tools in the proof is the decomposition of the functions in Morrey spaces based on the elliptic differential operators in question‎. ‎In some special cases where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood-Paley operator was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper are the decomposition results associated with elliptic differential operators and the result in the case where the explicit formula of the integral kernel of the heat semigroup is unavailable‎.

Vertical square functions and other operators associated with an elliptic operator

We study the vertical and conical square functions defined via elliptic operators in divergence form. In general, vertical and conical square functions are equivalent operators just in L2. But when this square functions are defined through the heat or Poisson semigroups that arise from an elliptic operator, we are able to find open intervals containing 2 where the equivalence holds. The intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off-diagonal estimates. As a consequence we obtain new boundedness results for some square functions. Besides, we consider a non-tangential maximal function associated with the Poisson semigroup and extend the known range where that operator is bounded. Our methods are based on the use of extrapolation for Muckenhoupt weights and change of angle estimates. All our results are obtained in the general setting of a degenerate elliptic operator, where the degeneracy is given by an A2 weight, in weighted Lebesgue spaces. Of course they are valid in the unweighted and/or non-degenerate situations, which can be seen as special cases, and they provide new results even in those particular settings.We also consider the square root of a degenerate elliptic operator in divergence form Lw and improve the lower bound of the interval where this operator is known to be bounded on Lp(vdw). Finally, we give unweighted boundedness results for the degenerate operators under consideration.

Gradient Estimates for the Commutator with Fractional Differentiation for Second Order Elliptic Perators

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 on ℝn. Let $$L^{\frac{\alpha}{2}}(0<\alpha<1)$$ denotes the fractional differential operator associated with L and $$(-\Delta)^{\frac{\alpha}{2}}b\;\in\;L^{n/\alpha}(\mathbb{R}^n)$$ . In this article, we prove that the commutator [b, $$L^{\frac{\alpha}{2}}$$ ] is bounded from the homogenous Sobolev space $$\dot{L}_\alpha^2$$ (ℝn) to L2(ℝn).

$L_p$ regularity theorem for elliptic equations in less smooth domains

We consider a $2m$th-order strongly elliptic operator $A$ subject to Dirichlet boundary conditions in a domain $\Omega$ of $\mathbb{R}^{n}$, and show the $L_{p}$ regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let $k$ be a nonnegative integer. When $A$ is a divergence form elliptic operator, $A-\lambda$ has a bounded inverse from the Sobolev space $W^{k-m}_{p}(\Omega)$ into $W^{k+m}_{p}(\Omega)$ for $\lambda$ belonging to a suitable sectorial region of the complex plane, if $\Omega$ is a uniformly $C^{k,1}$ domain. When $A$ is a non-divergence form elliptic operator, $A-\lambda$ has a bounded inverse from $W^{k}_{p}(\Omega)$ into $W^{k+2m}_{p}(\Omega)$, if $\Omega$ is a uniformly $C^{k+m,1}$ domain. Compared with the known results, we weaken the smoothness assumption on the boundary of $\Omega$ by $m-1$.

Green’s Function Estimates for Time-Fractional Evolution Equations

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.

Nondegenerate motion of singular points in obstacle problems with varying data

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points ([19]), and much older work by King, Lacey, and Vázquez gives an example of a singular free boundary point (in the Hele-Shaw flow) that remains stationary for a positive amount of time ([13]). The authors show how singular free boundaries in the obstacle problem in some settings move immediately in response to varying data. Three applications of this result are given, and in particular, the authors show a uniqueness result: For sufficiently smooth elliptic divergence form operators on domains in Rn and for the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct mean value sets (of the type found in [7] and [5]) which are centered at the same point do not intersect.

Trace operator and the Dirichlet problem for elliptic equations on arbitrary bounded open sets

We consider the Dirichlet problem and the weak Dirichlet problem on a general, possibly nonregular bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, integral, probabilistic, soft and the Perron-Wiener-Brelot solutions of the problem. We also study the relationship between solutions of the Dirichlet problem and weak Dirichlet problem. To this end, we extend the usual notion of the trace operator and consider the Dirichlet problem with Martin boundary.

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the Lp Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA0(x)∇ + B0(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space ℝ+n. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L = divA(x)∇ + B(x) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindos, Petermichl and Pipher.

Bounds on layer potentials with rough inputs for higher order elliptic equations

In this paper, we establish square-function estimates on the double and single layer potentials with rough inputs for divergence form elliptic operators, of arbitrary even order 2 m , with variable t-independent coefficients in the upper half-space.

Pointwise estimates of solutions to semilinear elliptic equations and inequalities

We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $\Omega$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.

Commutators with fractional differentiation for second-order elliptic operators on ℝn

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.