Nonlocal minimal surfaces Caffarelli, L.; Roquejoffre, J.-M.; Savin, O.
Communications on pure and applied mathematics,
September 2010, Letnik:
63, Številka:
9
Journal Article
We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions ...and on the fracture energy density, we show that minimizers are
C
1
,
1
/
2
, and that near non-degenerate points the fracture set is
C
1
,
α
, for some
α
∈
(
0
,
1
)
.
Here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems. The limiting problems involve multiple valued harmonic functions or, in general, harmonic maps to ...singular spaces and free interfaces between supports of various components of the maps. The main results of the paper are the uniform Lipschitz regularity of solutions as well as the regularity of free interfaces.
We study the regularity of the "free surface" in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a${\rm{C}}^{1,\alpha } (n - 2)$-dimensional surface ...in${\rm{R}}^{n - 1} $.
In this paper we give a counter-example to the homogenization of the forced mean curvature motion in a periodic setting in dimension
N
≧
3
when the forcing is positive. We also prove a general ...homogenization result for geometric motions in dimension
N
= 2 under the assumption that there exists a constant δ > 0 such that every straight line moving with a normal velocity equal to δ is a subsolution for the motion. We also present a generalization in dimension 2, where we allow sign changing normal velocity and still construct bounded correctors, when there exists a subsolution with compact support expanding in all directions.
Obstacle-type problems for minimal surfaces Caffarelli, L.; De Silva, D.; Savin, O.
Communications in partial differential equations,
08/2016, Letnik:
41, Številka:
8
Journal Article
Recenzirano
We study certain obstacle-type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.
Our goal in this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half-plane in which reaction-diffusion phenomena occur exchanges mass with a ...line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi, and the second author as a model of enhancement of biological invasions by a line of fast diffusion. It was observed numerically by A.-C. Coulon that the leading edge of the front, rather than being located on the line, was in the lower half-plane.
We explain this behavior for a closely related free boundary problem. We construct travelling waves for this problem, and the analysis of their free boundary near the line confirms the predictions of the numerical simulations.
<< Here, the C^\alpha regularity of such weak solutions is established in the difficult fractional exponent case s=1/2. For the other fractional exponents s\in (0,1) this H>ö
Random homogenization of an obstacle problem Caffarelli, L.A.; Mellet, A.
Annales de l'Institut Henri Poincaré. Analyse non linéaire,
03/2009, Letnik:
26, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We study the homogenization of an obstacle problem in a perforated domain, when the holes are periodically distributed and have random shape and size. The main assumption concerns the capacity of the ...holes which is assumed to be stationary ergodic.
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