Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

TL;DR

This paper proves well-posedness and high regularity for the heat equation with Robin boundary conditions in a 2D wedge, using weighted spaces, without restrictions on the wedge's opening angle.

Contribution

It develops a new mathematical framework that achieves high regularity results without smallness assumptions on the wedge's opening angle, addressing scaling invariance issues.

Findings

Established well-posedness in weighted L^2 spaces.

Achieved arbitrarily high regularity without small angle restrictions.

Handled scaling invariance breakings in the resolvent problem.

Abstract

Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.