A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic domains in $R^d$

Abstract

We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains D of the typeWe obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.

Sobolev space theory and Hölder estimates for the stochastic partial differential equations on conic and polygonal domains

“…We then set a program of three steps: (i) preparing a refined d-dimensional Green's function estimate for operators with measurable coefficients (ii) preparing PDE result (iii) establishing SPDE result addressing the higher order derivative estimates. First two steps are done in [7] and [8], and this article fulfills the last step. In [7] the refined Green's function estimate involves both the distance to the vertex and the distance to the boundary and it now vanishes at all the points on the boundary with informative decay rate near the boundary.…”

“…In [7] the refined Green's function estimate involves both the distance to the vertex and the distance to the boundary and it now vanishes at all the points on the boundary with informative decay rate near the boundary. The work [8] fully makes use of what we prepared in [7] and it is designed to serve this article well. For instance, in this article we frequently consider the difference of two slightly different SPDEs, which yields a PDE and then the results in [8] intervene.…”

“…The work [8] fully makes use of what we prepared in [7] and it is designed to serve this article well. For instance, in this article we frequently consider the difference of two slightly different SPDEs, which yields a PDE and then the results in [8] intervene. To make [8] wellprepared for SPDEs, it is important for us to embrace both the strategy using equation itself and the one using a solution representation with Green's function.…”

“…Let us explain, in details, the work done in [8] and the role of it in relation with this article. In [8] we studied the (deterministic) parabolic equation on conic domains, using a mixed weight system involving both the distance to the vertex and the boundary. Now, we have two different ranges of powers of the distances.…”

“…Now, we have two different ranges of powers of the distances. In [8] we aim that the range for the power of the distance to the boundary is the same as the C 1 -domain result and the one for the power of the distance to the vertex embraces the results of the model case in [3] and [2]. The weight system in [8] is more natural and appropriate compared to the one in [3] and [2], and leads us to a significant improvement of the result therein at least with PDEs.…”