A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains

I. Smears and E. Süli designed and analyzed a discontinuous Galerkin finite element method for the approximation of solutions to elliptic partial differential equations in nondivergence form. The results were proven, based on the assumption that the computational domain was convex and polytopal. In this paper, we extend this framework, allowing for Lipschitz continuous domains with piecewise curved boundaries.     

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.     

Two-sided and one-sided Liouville theorems for quasilinear elliptic equations of the nondivergence form

One gives the development of the author's previous results regarding two-sided Liouville theorems and one proves some one-sided Liouville theorems for some classes of elliptic second-order equations of the nondivergence form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 59–80, 1980.     

Harmonic measures for elliptic operators of nondivergence form

Given any in (n–2, n–1), there exists a uniformly elliptic operator of nondivergence form in the upper half space ₊ ⁿ , so that the corresponding harmonic measure is supported on a set of Hausdorff dimension at most .Partially supported by the National Science Foundation     

Blow-up of solutions for a class of nondivergence elliptic inequalities

Blow-up conditions are obtained for second-order nondivergence elliptic inequalities containing terms with lower order derivatives.     

A higher integrability result for nondivergence elliptic equations

The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type . We assume that the coefficients a _ij are bounded and have small BMO-norm.      

Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations

We show that theL ^p norms, 0<p<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectivelyA _∞ with respect to the corresponding harmonic measure or normalized harmonic measure. Both authors are supported by NSF     

Nontrivial solvability of elliptic equations in divergence form with complex coefficients

The stationary Fokker-Planck-Kolmogorov equation with complex diffusion coefficients and a complex vector-field is examined on a torus. Under suitable conditions for the diffusion coefficients, it is proven that a nontrivial solution exists and the solution space is multidimensional in some cases.     

Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth

In this paper we are concerned with the regularity in Morrey spaces for weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth condition. Interior Hölder continuity of weak solutions is also derived with the improvement of the given data regularities.     

Generalized solutions for a class of non-uniformaly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence from with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV ^loc and are C ^1,α almost everywhere.     

Global weighted estimates for second-order nondivergence elliptic and parabolic equations

In this paper, we obtain the global weighted \(L^p\) estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain \(L^p\)-type regularity estimates for such equations.