Operators In Non-divergence Form Research Articles

On the Kato Problem for Elliptic Operators in Non-Divergence Form

On the Kato Problem for Elliptic Operators in Non-Divergence 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

Note on Green’s functions of non-divergence elliptic operators with continuous coefficients

We improve a result in Kim and Lee [Ann. Appl. Math. 37 (2021), pp. 111–130], showing that if the coefficients of an elliptic operator in non-divergence form are of Dini mean oscillation, then its Green’s function has the same asymptotic behavior near the pole x 0 x_0 as that of the corresponding Green’s function for the elliptic equation with constant coefficients frozen at x 0 x_0 .

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

On the boundary Harnack principle in Hölder domains

<abstract> We investigate the boundary Harnack principle for uniformly elliptic operators in divergence form in Hölder domains of exponent $ \alpha &gt; 0 $. We also deal with operators in nondivergence form with coefficient that remain constant in the graph direction. </abstract>

Harnack inequality for nonlocal operators on manifolds with nonnegative curvature

We establish the Krylov–Safonov Harnack inequalities and Hölder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity estimates for these operators which recover the classical estimates for second order local operators as limits.

New boundary Harnack inequalities with right hand side

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in Lq with q>n. Our approach is based on the scaling and comparison arguments of [13], and we show that all our assumptions are sharp.As a consequence of our results, we deduce the C1,α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

On certain degenerate and singular elliptic PDEs III: Nondivergence form operators and RH∞-weights

A Harnack inequality is proved for nonnegative strong solutions to certain second-order elliptic PDEs with rough coefficients whose degeneracies/singularities are shaped by weights in the reverse-Hölder class RH∞. Such operators include subelliptic Grushin and Grushin-like PDEs with unbounded first-order terms.

The obstacle problem for a class of degenerate fully nonlinear operators

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: $$ \left{\begin{array}{rll} \min\left{f-|Du|^\gamma F(D^2u),u-\phi\right} &= 0 & \textrm{ in } \Omega,\ u & = g & \textrm{ on } \partial \Omega, \end{array}\right. $$ for some degeneracy parameter $\gamma\geq 0$, uniformly elliptic operator $F$, bounded source term $f$, and suitably smooth obstacle $\phi$ and boundary datum $g$. We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely $\partial{u>\phi} \cap \Omega$. In particular, for the homogeneous case ($f\equiv0$) we get that solutions are $C^{1,1}$ at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also prove several non-degeneracy properties of solutions and partial results regarding the free boundary. These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler prototype given by an operator of the form $\mathcal{G}\[u] = |Du|^\gamma\Delta u$, with $\gamma >0$ and $f \equiv 1$.

An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

A maximum principle for non-negative zeroth order coefficient in some unbounded domains

We study a maximum principle for a uniformly elliptic second order differential operator in nondivergence form. We allow a bounded positive zeroth order coefficient in a certain type of unbounded domains. The results extend a result by J. Busca in a sense of domains, and we present a simple proof based on local maximum principle by Gilbarg and Trudinger with iterations.

Numerical Simulation of Two-Dimensional Pucci’s Equation with Dirichlet Boundary Conditions Using Nonvariational Finite Element Method

This paper provides numerical simulation of two-dimensional Pucci’s equation with Dirichlet boundary conditions. The Pucci’s operator is a prototype of a nonlinear operator in non-divergence form. Non-divergence form makes numerical solution challenging because standard tools like Pohozaev identity and global integration by parts are no longer applicable. Therefore, present study for the numerical solution relies on non-variational finite element method, which is independent of above said tools. After non-variational space discretization the resulting finite dimensional problem is solved by Newton iterative method.

The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

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 Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 ∖Ω, and u1,u2 ≥ 0 in ℝn, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Holder continuous up to the boundary in B1/2. These results will be used in forthcoming works on obstacle-type problems for nonlocal operators.

On a non-smooth potential analysis for Hörmander-type operators

We develop a potential theory approach for some degenerate parabolic operators in non-divergence form and with non-smooth coefficients, which are modeled on smooth Hormander vector fields. We prove necessary and sufficient Wiener-type tests for the regularity of boundary points. As a consequence we obtain, in particular, a cone-type criterion. We also investigate the related boundary value problem and the Holder regularity at the boundary.

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

ABSTRACTWe prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.

On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations

We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N.Q. Le.Since our results are shown to apply to the convex functions |x|p with p≥2 and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot–Carathéodory metric in the study of certain subelliptic PDEs.

On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.

Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains

We study a Venttsel' problem in a three dimensional fractal domain for an operator in non divergence form. We prove existence, uniqueness and regularity results of the strict solution for both the fractal and prefractal problem, via a semigroup approach. In view of numerical approximations, we study the asymptotic behaviour of the solutions of the prefractal problems and we prove that the prefractal solutions converge in the Mosco-Kuwae-Shioya sense to the (limit) solution of the fractal one.

Semigroups generated by elliptic operators in non-divergence form on C-o(Ω)

Let  ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = Xn i, j=1 ai j @i j u + Xn j=1 b j @ j u + cu where a j i = ai j 2 C(¯ ) and b j , c 2 L1(), c  0 . We show that the realization A0 of A in C0() := {u 2 C(¯ ) : u|@ = 0} given by D(A0) := {u 2 C0() W2,n loc () : Au 2 C0()} A0u := Au generates a bounded holomorphic C0-semigroup on C0(). The result is in particular true if  is a Lipschitz domain. So far the best known result seems to be the case where  has C2-boundary [12, Section 3.1.5]. We also study the elliptic problem −Au = f u|@ = g .

The Caffarelli Alternative in Measure for the Nondivergence Form Elliptic Obstacle Problem with Principal Coefficients in VMO

We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in [3].