Smoothness Properties of Bounded Solutions of Dirichlet's Problem for Elliptic Equations in Regions with Corners on the Boundary

Abstract

We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

Multiscale finite element for problems with highly oscillatory coefficients

“…. , n) under the compatibility conditions stated in [4]. Under the assumptions (3.21), for convex polygonal domains we have…”

On the shape of the free boundary of variational inequalities with gradient constraints

“…It is easy to see that if x ∈ P and y ∈ ∂U is one of the p-closest points to x on the boundary, then the segment between x and y (which is obviously in U ) lies inside P . In addition, we have ∆u = −η over E, 1 and ∆u ≥ −η a.e. over U .…”

“…As nonreentrant corners have an elastic neighborhood in U , around them we have ∆u = −η. Now as u vanishes on ∂U , we can apply the results in Azzam [1] to deduce that u is in C 1,α for some α > 0 around these corners.…”

A new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][16][17][18][19][20][21][22][23]) can be used to solve all kinds of second order elliptic problems with rapidly oscillating coefficients very effectively, for it couples the macroscopic and microscopic scales together.…”