Elliptic Equations In Divergence Form Research Articles

Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.

Critical sets of solutions of elliptic equations in periodic homogenization

AbstractIn this paper we study critical sets of solutions of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the ‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for solutions are bounded. The key step is an estimate of “turning” of an approximate tangent map, the projection of a non‐constant solution onto the subspace of spherical harmonics of order ℓ, when the doubling index for on a sphere is trapped between and , for r between 1 and a minimal radius . This estimate is proved by using harmonic approximation successively. With a suitable L2 renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

Matrix weights and regularity for degenerate elliptic equations

We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.

Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf–Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf–Oleinik lemma.

Inverse conductivity Problem with Internal Data

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

Locality properties of standard homogenization commutator

We study stochastic homogenization for linear elliptic equations in divergence form and focus on the recently developed theory of fluctuations. It has been observed that the fluctuations of averages of the solution are captured by the so-called standard homogenization commutator Ξ ε o . Our aim is to study how Ξ ε o (and its higher-order analogs) decorrelates on large scales when averaged on balls which are far enough. Taking advantage of its approximate locality, we give a quantitative characterization of this decorrelation in terms of both the macroscopic scale and the distance between the balls showing that Ξ ε o inherits the correlation properties of the environment.

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

Isogeometric Analysis Approximation of Linear Elliptic Equations with L1 Data

Isogeometric Analysis (IgA) is a recent technique for the discretization of Partial Differential Equations (PDEs). The main feature of the method is the ability to maintain the same exact description of the computational domain geometry throughout the analysis process, including refinement. In the present paper, we consider, in dimension d >= 2 the Isogeometric Analysis approximation of second order elliptic equations in divergence form with right-hand side in L1 . We assume that the family of meshes is shape regular and satisfies the discrete maximum principle. When the right-hand side belongs to L1(\Omega), we prove that the unique solution of the discrete problem converges to the unique renormalized solution in W01,q(\Omega), 1 <= q < d/(d-1) . We also prove some error estimates and include numerical tests for data with low smoothness.

Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

On divergence-free (form-bounded type) drifts

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.

The Boyarsky–Meyers Estimate for Second Order Elliptic Equations in Divergence Form. Two Spatial Examples

The Boyarsky–Meyers Estimate for Second Order Elliptic Equations in Divergence Form. Two Spatial Examples

Zvonkin’s transform and the regularity of solutions to double divergence form elliptic equations

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to some power p > d. We establish new estimates for the Lp -norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini’s condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin’s transform of the drift coefficient.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

Maximal L1 regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

This paper is devoted to the maximal L1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L1-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal L1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids.

Lipschitz bounds for integral functionals with (p,q)-growth conditions

Abstract We study local regularity properties of local minimizers of scalar integral functionals of the form ℱ ⁢ [ u ] := ∫ Ω F ⁢ ( ∇ ⁡ u ) - f ⁢ u ⁢ d ⁢ x \mathcal{F}[u]:=\int_{\Omega}F(\nabla u)-fu\,dx where the convex integrand F satisfies controlled ( p , q ) {(p,q)} -growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term f and improved assumptions on the growth conditions on F with respect to the existing literature. Along the way, we establish an L ∞ {L^{\infty}} - L 2 {L^{2}} -estimate for solutions of linear uniformly elliptic equations in divergence form, which is optimal with respect to the ellipticity ratio of the coefficients.

Boundary regularity estimates in Hölder spaces with variable exponent

We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in Hölder spaces with variable exponent. The procedure allows to extend the estimates up to a portion of the boundary where Dirichlet or Neumann boundary conditions are prescribed and produces a Schauder theory for partial derivatives of solutions of any order kin {mathbb {N}}. The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The given data of the problem are taken in Hölder and Lebesgue spaces, both with variable exponent.

Boundary Lipschitz regularity of solutions for general semilinear elliptic equations in divergence form

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.

The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization

This article is about the quantitative homogenization theory of linear elliptic equations in divergence form with random coefficients. We derive gradient estimates on the homogenization error, i. e. on the difference between the actual solution and the two-scale expansion of the homogenized solution, both in terms of strong norms (oscillation) and weak norms (fluctuation). These estimates are optimal in terms of scaling in the ratio between the microscopic and the macroscopic scale.The purpose of this article is to highlight the usage of the recently introduced annealed Calderón-Zygmund (CZ) estimates in obtaining the above, previously known, error estimates. Moreover, the article provides a novel proof of these annealed CZ estimate that completely avoids quenched regularity theory, but rather relies on functional analysis. It is based on the observation that even on the level of operator norms, the Helmholtz projection is close to the one for the homogenized coefficient (for which annealed CZ estimates are easily obtained).In this article, we strive for simple proofs, and thus restrict ourselves to ensembles of coefficient fields that are local transformations of Gaussian random fields with integrable correlations and Hölder continuous realizations. As in earlier work, we use the natural objects from the general theory of homogenization, like the (potential and flux) correctors and the homogenization commutator. Both oscillation and fluctuation estimates rely on a sensitivity calculus, i.e. on estimating how sensitively the quantity of interest does depend on an infinitesimal change in the coefficient field, which is fed into the Spectral Gap inequality. In this article, the annealed CZ estimate is the only form in which elliptic regularity theory enters.

Vortex patch problem for steady lake equation

We study the vortex patch problem for the steady lake equation in a bounded domain and construct three different kinds of solutions where the vorticity concentrates in the domain or near the boundary. Our approach is based on the Lyapunov–Schmidt reduction, which transforms the construction into a problem of seeking critical points for a function related to the kinetic energy. The method in this paper has a wide applicability and can be used to deal with general elliptic equations in divergence form with Heaviside nonlinearity.