Kyeonghun Kim — Page 2

arXiv2021

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

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 type $$ D(M):=\left\{x\in R^d :\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We 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.

math.AP↗

arXiv2017

An lp-boundedness of stochastic singular integral operators and its application to spdes

In this article we introduce a stochastic counterpart of the Hörmander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $ρ$ on $(0,\infty)\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty) \times R^d$, $$ \sup_{X,Y}\int_{0}^\infty \left[ \int_{ρ(X,Z) \geq C_0 ρ(X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right]^2 dr <\infty. $$ We prove that the stochastic singular integral of the type $$ \mathbb{T} g(t,x) :=\int_0^{t} \int_{R^d} K(t,s,x,y) g(s,y)dy dW_s $$ is a bounded operator on $\mathbb{L}_p=L_p(Ω\times (0,\infty); L_{p}(R^d))$ for any $p\geq 2$ if it is bounded when $p=2$ and stochastic Hörmander condition holds. Here $Ω$ is a probability space and $W_t$ is a Wiener process on $Ω$. Proving the $L_p$-boundedness of such integral operators is the key step in constructing an $L_p$-theory for linear stochastic partial differential equations (SPDEs in short). As a byproduct of our result on stochastic singular operators we obtain the maximal $L_p$-regularity result for a very wide class of SPDEs.

math.PR↗