Regularity for a clamped grid equation $u_{xxxx}+u_{yyyy}=f $ on a domain with a corner

Abstract

The operator $L=frac{partial ^{4}}{partial x^{4}} +frac{partial ^{4}}{partial y^{4}}$ appears in a model for the vertical displacement of a two-dimensional grid that consists of two perpendicular sets of elastic fibers or rods. We are interested in the behaviour of such a grid that is clamped at the boundary and more specifically near a corner of the domain. Kondratiev supplied the appropriate setting in the sense of Sobolev type spaces tailored to find the optimal regularity. Inspired by the Laplacian and the Bilaplacian models one expect, except maybe for some special angles that the optimal regularity improves when angle decreases. For the homogeneous Dirichlet problem with this special non-isotropic fourth order operator such a result does not hold true. We will show the existence of an interval $( frac{1}{2}pi ,omega _{star })$, $omega _{star }/pi approx 0.528dots$ (in degrees $omega _{star }approx 95.1dots^{circ} $), in which the optimal regularity improves with increasing opening angle.

Keywords

• Nonisotropic
• fourth order PDE
• domain with corner
• clamped grid
• weighted Sobolev space
• regularity