868 research outputs found
Homogenization of the Oscillating Dirichlet Boundary Condition in General Domains
We prove the homogenization of the Dirichlet problem for fully nonlinear
elliptic operators with periodic oscillation in the operator and of the
boundary condition for a general class of smooth bounded domains. This extends
the previous results of Barles and Mironescu in half spaces. We show that
homogenization holds despite a possible lack of continuity in the homogenized
boundary data. The proof is based on a comparison principle with partial
Dirichlet boundary data which is of independent interest.Comment: Version to appear in J. Math. Pures Appl. Added Remarks 1.2 and 1.7.
Removed some extraneous statements of previous results (previously
Corollaries 2.7 and 2.12). Changed the statement and proof of Lemma 3.1 to
fix a small error and R^{-\alpha} is now (N/R)^{\alpha} here and in later
uses of the Lemma. 23 page
Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
We study the averaging behavior of nonlinear uniformly elliptic partial
differential equations with random Dirichlet or Neumann boundary data
oscillating on a small scale. Under conditions on the operator, the data and
the random media leading to concentration of measure, we prove an almost sure
and local uniform homogenization result with a rate of convergence in
probability
An L1 Penalty Method for General Obstacle Problems
We construct an efficient numerical scheme for solving obstacle problems in
divergence form. The numerical method is based on a reformulation of the
obstacle in terms of an L1-like penalty on the variational problem. The
reformulation is an exact regularizer in the sense that for large (but finite)
penalty parameter, we recover the exact solution. Our formulation is applied to
classical elliptic obstacle problems as well as some related free boundary
problems, for example the two-phase membrane problem and the Hele-Shaw model.
One advantage of the proposed method is that the free boundary inherent in the
obstacle problem arises naturally in our energy minimization without any need
for problem specific or complicated discretization. In addition, our scheme
also works for nonlinear variational inequalities arising from convex
minimization problems.Comment: 20 pages, 18 figure
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