Elliptic Equations In Nondivergence Form Research Articles

Interior W2,p(x)-regularity for elliptic equations of nondivergence form

In this paper, we consider the following elliptic equations of nondivergence form∑i,j=1naij∂2u∂xi∂xj=f,in Ω⊂Rn, where the coefficients aij are real valued bounded measurable functions defined in Ω⊂Rn, belong to VMOloc (see Definition 2.5) and satisfy the uniformly elliptic condition. Under the assumption that p(⋅) satisfies the log-Hölder continuity condition, we will establish the interior W2,p(x)-regularity for strong solutions to the elliptic equations of nondivergence form.

Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.

$$W^{2,p(\cdot )}$$ W 2 , p ( · ) -regularity for elliptic equations in nondivergence form with BMO coefficients

We establish an optimal $$W^{2,p(\cdot )}$$ -estimate to the Dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a $$C^{1,1}$$ bounded domain for every variable exponent $$p(\cdot )$$ with log-Holder continuity. The matrix of the coefficients is assumed to have a small BMO semi-norm, depending on the exponent, the boundary of the domain, and the matrix itself.

Generalized Harnack Inequality for Nonhomogeneous Elliptic Equations

This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack's inequality. This paper presents a new type of generalization of the classical Harnack's inequality for such equations. As a corollary we obtain the optimal Harnack type of estimate for p(x)-harmonic functions which quantifies the strong minimum principle.

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by −κ for κ≥0. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge-Amp\`ere equation.

Boundary behavior of solutions of elliptic equations in nondivergence form

A new proof for boundary Lipschitz estimate is shown for solutions of elliptic equations in nondivergence form. Under additional condition, the boundary differentiability is obtained.

Estimates for nondivergence elliptic equations with VMO coefficients in generalized grand Morrey spaces

In this paper, we establish the regularity, in generalized grand Morrey spaces, of the solution to elliptic equations in non-divergence form with VMO coefficients by means of the theory of singular integrals and linear commutators.

Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited

We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In the latter case, we overcome difficulties caused by a lack of estimates for the first derivatives of approximate correctors by modifying the perturbed test function argument to take advantage of the spreading of the contact set.

Discontinuous Galerkin Finite Element Approximation of Nondivergence Form Elliptic Equations with Cordès Coefficients

Nondivergence form elliptic equations with discontinuous coefficients do not generally possess a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. We propose a new $hp$-version discontinuous Galerkin finite element method for a class of these problems which satisfy the Cordès condition. It is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size $h$ and suboptimal with respect to the polynomial degree $p$ by only half an order. Numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under $hp$-refinement for problems with discontinuous coefficients and nonsmooth solutions.

Weighted Herz Spaces and Regularity Results

It is proved that, for the nondivergence form elliptic equations <svg style="vertical-align:-6.50204pt;width:114.9625px;" id="M1" height="20.85" version="1.1" viewBox="0 0 114.9625 20.85" width="114.9625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.85)"> <g transform="translate(72,-55.32)"> <text transform="matrix(1,0,0,-1,-71.95,61.83)"> <tspan style="font-size: 12.50px; " x="0" y="0">∑</tspan> </text> <text transform="matrix(1,0,0,-1,-60.52,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> <tspan style="font-size: 8.75px; " x="0" y="9.9300003">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="9.9300003">,</tspan> <tspan style="font-size: 8.75px; " x="4.9098721" y="9.9300003">𝑗</tspan> <tspan style="font-size: 8.75px; " x="8.3143997" y="9.9300003">=</tspan> <tspan style="font-size: 8.75px; " x="14.30952" y="9.9300003">1</tspan> </text> <text transform="matrix(1,0,0,-1,-39.25,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan> </text> <text transform="matrix(1,0,0,-1,-32.98,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-25.83,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-19.9,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-15.09,56.54)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑖</tspan> </text> <text transform="matrix(1,0,0,-1,-12.65,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-7.83,56.54)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-0.56,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="12.040389" y="0">𝑓</tspan> </text> </g> </g> </svg>, if <svg style="vertical-align:-2.34499pt;width:10.675px;" id="M2" height="13.4875" version="1.1" viewBox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑓</tspan> </text> </g> </g> </svg> belongs to the weighted Herz spaces <svg style="vertical-align:-4.27347pt;width:58.799999px;" id="M3" height="17.9" version="1.1" viewBox="0 0 58.799999 17.9" width="58.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.9)"> <g transform="translate(72,-57.68)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-61.42,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>, then <svg style="vertical-align:-6.50204pt;width:105.4375px;" id="M4" height="20.6875" version="1.1" viewBox="0 0 105.4375 20.6875" width="105.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.6875)"> <g transform="translate(72,-55.45)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-66.02,58.86)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-61.21,56.67)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑖</tspan> </text> <text transform="matrix(1,0,0,-1,-58.77,58.86)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-53.95,56.67)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-46.68,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">∈</tspan> <tspan style="font-size: 12.50px; " x="12.027886" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-24.11,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-19.26,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>, where <svg style="vertical-align:-0.11285pt;width:7.5374999px;" id="M5" height="7.1624999" version="1.1" viewBox="0 0 7.5374999 7.1624999" width="7.5374999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> </g> </g> </svg> is the <svg style="vertical-align:-0.20064pt;width:32.137501px;" id="M6" height="14.025" version="1.1" viewBox="0 0 32.137501 14.025" width="32.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.025)"> <g transform="translate(72,-60.78)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑊</tspan> </text> <text transform="matrix(1,0,0,-1,-57.68,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0">,</tspan> <tspan style="font-size: 8.75px; " x="6.5640001" y="0">𝑝</tspan> </text> </g> </g> </svg>-solution of the equations. In order to obtain this, the authors first establish the weighted boundedness for the commutators of some singular integral operators on <svg style="vertical-align:-4.27347pt;width:58.799999px;" id="M7" height="17.9" version="1.1" viewBox="0 0 58.799999 17.9" width="58.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.9)"> <g transform="translate(72,-57.68)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-61.42,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>.

Second-Order Elliptic Equations of Nondivergence Form with Small BMO Coefficients in ℝ n

In this paper we obtain the global regularity estimates in Orlicz spaces for second-order elliptic equations of nondivergence form with small BMO coefficients in ℝn. As a corollary we obtain Lp-type regularity for such equations. Our results improve the known results for such problems.

Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman–Isaacs equations from stochastic control problems. We establish an Alexandroff–Bakelman–Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.

Second-order elliptic and parabolic equations with $B(\mathbb{R}^{2}, VMO)$ coefficients

The solvability in Sobolev spaces W 1,2 p is proved for nondivergence form second-order parabolic equations for p > 2 close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the W 2 p -solvability for the same p of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when p = 2.

Removable singularities for solutions of second-order linear uniformly elliptic equations in non-divergence form

Let be a linear uniformly elliptic operator of the second order in , , with bounded measurable real coefficients, that satisfies the weak uniqueness property. The removability of compact subsets of a domain is studied for weak solutions of the equation (in the sense of Krylov and Safonov) in some classes of continuous functions in . In particular, a metric criterion for removability in Hölder classes with small exponent of smoothness is obtained. Bibliography: 20 titles.

Maximum Principles for Second Order Elliptic Equations in Nondivergence Form and Applications

The main objectives of this study were to introduce a maximum principle for second order elliptic equations in nondivergence form and to use a nondivergence technique called the ABP method to establish the maximum principles in small domains. The advantage of this work was that we obtain a maximum principle for equations with discontinuous coefficients of the leading terms and the ABP estimate was a bound for solutions of uniformly elliptic equations written in nondivergence form with measurable coefficients. It was an essential tool in the regularity theory for fully nonlinear equations.

On some results in weighted spaces under Cordes type conditions

We study Dirichlet problem for linear elliptic equations in nondivergence form with discontinuous coefficients when the class of discontinuity is of Cordes type and domains are unbounded. In particular we state some local and non local a priori bounds in weighted spaces where the weight is related to the distance function from a fixed subset S of ¶­ and study the dependence of the constants in the estimates. The coefficients of lower terms in the differential operator belong to weighted spaces and the principal coefficients are ‘near’ to functions satisfying a condition of Cordes type. This condition allow us to apply embedding results near to a subset of ¶­ and for |x| large enough without further assumptions.

Elliptic Differential Equations with Coefficients Measurable with Respect to One Variable and VMO with Respect to the Others

We prove the unique solvability of second order elliptic equations in nondivergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the Martingale problem associated with the elliptic equations.

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u0 such that the linearized at u0 problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u0 which depends smoothly (in W2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W2,p) solutions for u0.