arXiv subjects
Publications and source records attributed to Kyeong-Hun Kim.
We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.
This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized Gaussian noise. Our main results include the existence and uniqueness of strong solutions in weighted Sobolev spaces, along with maximal $L_p$-regularity estimates for the solutions.
We provide Green's function estimates for parabolic operators on polyhedrons and polyhedral cones in $\mathbb{R}^3$. These estimates incorporate mixed weights, which include appropriate powers of the distances to the vertices, the edges, and the boundary of the domains. The allowable ranges for the weight parameters are explicitly determined by the geometry of the domains.
We introduce a weighted Sobolev space theory for the non-local elliptic equation $$ \Delta^{\alpha/2}u=f, \quad x\in \mathcal{O}\,; \quad r_{\overline{\mathcal{O}}^c}u=g $$ as well as for the non-local parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in \mathcal{O} \,; \quad r_{\mathcal{O}}u(0,\cdot)=u_0, \,r_{(0,T)\times \overline{\mathcal{O}}^c}u=g. $$ Here, $\alpha\in (0,2)$ and $\mathcal{O}$ is a $C^{1,1}$ open set. We prove uniqueness and existence results in weighted Sobolev spaces. We measure the Sobolev and H\"older regularities of arbitrary order derivatives of solutions using a system of weights consisting of appropriate powers of the distance to the boundary. One of the most interesting features of our results is that, unlike the classical results in Sobolev spaces without weights, the weighted regularities of solutions in $\mathcal{O}$ are less affected by those of exterior conditions on $\overline{\mathcal{O}}^c$. For instance, even if $g=\delta_{x_0}$, the dirac delta distribution concentrated at $x_0\in \overline{\mathcal{O}}^c $, the solution to the elliptic equation given with $f=0$ is infinitely differentiable in $\mathcal{O}$, and for any $k=0,1,2, 3,\cdots$, $\varepsilon>0$, and $\delta\in (0,1)$, it holds that $$ |d_x^{-\frac{\alpha}{2}+\varepsilon+k}D^k_xu|_{C_b(\mathcal{O})} +|d_x^{-\frac{\alpha}{2}+\varepsilon+k+\delta} D^k_xu|_{C^{\delta}(\mathcal{O})}<\infty, $$ where $d_x=dist(x, \partial \mathcal{O})$.
We study the zero exterior problem for the elliptic equation $$ Δ^{α/2}u-λu=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=Δ^{α/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $α\in (0,2)$, $λ\geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.
We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation $$ du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t, \quad t>0, \,x\in \mathcal{D} $$ given with non-zero initial data. Here $\{w^k_t: k=1,2,\cdots\}$ is a family of independent Wiener processes defined on a probability space $(Ω, \mathbb{P})$, $a^{ij}=a^{ij}(ω,t)$ are merely measurable functions on $Ω\times (0,\infty)$, and $\mathcal{D}$ is either a polygonal domain in $\mathbb{R}^2$ or an arbitrary dimensional conic domain of the type \begin{equation} \label{conic} \mathcal{D}(\mathcal{M}):=\left\{x\in \mathbb{R}^d :\,\frac{x}{|x|}\in \mathcal{M}\right\}, \quad \quad \mathcal{M}\in S^{d-1}, \quad (d\geq 2) \end{equation} where $\mathcal{M}$ is an open subset of $S^{d-1}$ with $C^2$ boundary. We measure the Sobolev and Hölder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.
We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $$ \partial_{t}^αu=\left( ϕ(Δ) u +f(u) \right) + \partial_t^β\sum_{k=1}^\infty \int_0^t g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0 $$ as well as the SPDE driven by space-time white noise $$ \partial^α_{t}u=ϕ(Δ)u + f(u) + \partial^{β-1}_{t}h(u) \dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}. $$ Here, $α\in (0,1), β\in (-\infty, α+1/2)$, $\{w_t^k : k=1,2,\cdots\}$ is a family of independent one-dimensional Wiener processes, and $\dot{W}$ is a space-time white noise defined on $[0,\infty)\times \mathbb{R}^d$. The time non-local operator $\partial_{t}^γ$ denotes the Caputo fractional derivative if $γ>0$ and the Riemann-Liouville fractional integral if $γ\leq0$. The the spatial non-local operator $ϕ(Δ)$ is a type of integro-differential operator whose symbol is $-ϕ(|ξ|^2)$, where $ϕ$ is a Bernstein function satisfying \begin{equation*} κ_0\left(\frac{R}{r}\right)^{δ_{0}} \leq \frac{ϕ(R)}{ϕ(r)}, \qquad \forall\,\, 0 0$ and $δ_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.
We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by Lévy processes of the type $$ \partial^α_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^β_{t}\int_{0}^{t} (\sum_{i=1}^dμ^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^α_t$ and $\partial^β_t$ are the Caputo fractional derivatives, $α\in (0,2), β\in (0,α+1/p)$, and $\{Z^k_t:k=1,2,\cdots\}$ is a sequence of independent Lévy processes. The coefficients are random functions depending on $(t,x)$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity of the solution.
In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.
We present weighted Sobolev spaces and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.
We present a refined Green's function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.
We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with Muckenhoupt weights for the equation $$ \partial_{t}^αu(t,x)=Δu(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $α\in (0,2)$ and $\partial_{t}^α$ is the Caputo fractional derivative of order $α$. In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and $w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N \int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$ weights. Our approach is based on the sharp function estimates of the derivatives of solutions.
We establish existence, uniqueness and higher order weighted $L_p$-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\mathbb{R}^2$. We use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable. In this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand. The range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp.
Let $X=(X_t)_{t \ge 0}$ be a stochastic process which has an (not necessarily stationary) independent increment on a probability space $(Ω, \mathbb{P})$. In this paper, we study the following Cauchy problem related to the stochastic process $X$: $\label{main eqn} \frac{\partial u}{\partial t}(t,x) = \cA(t)u(t,x) +f(t,x), \quad u(0,\cdot)=0, \quad (t,x) \in (0,T) \times \mathbf{R}^d, \end{align} where $f \in L_p( (0,T) ; L_p(\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \begin{align*} \cA(t)u(t,x) = \lim_{h \downarrow 0}\frac{\mathbb{E}\left[u(t,x+X_{t+h}-X_t)-u(t,x)\right]}{h}$. We provide a sufficient condition on $X$ to guarantee the unique solvability of equation (\ref{ab main}) in $L_p\left( [0,T] ; H^ϕ_{p}\right)$, where $H^ϕ_{p}$ is a $ϕ$-potential space on $\mathbf{R}^d$ . Furthemore we show that for this solution, \| u\|_{L_p\left( [0,T] ; H^ϕ_{p}\right)} \leq N \|f\|_{L_p\left( [0,T] ; L_p\right)}, where $N$ is independent of $u$ and $f$.
We prove a weighted $L_p$-estimate for the stochastic convolution associated to the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain $\mathcal{D}_{κ_0}\subset\mathbb{R}^2$ with angle $κ_0\in(0,2π)$. Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted $L_p$-Sobolev spaces. In order to capture the singular behaviour of the solution and its derivatives at the vertex, we use powers of the distance to the vertex as weight functions. The admissible range of weight parameters depends explicitly on the angle $κ_0$.
Let $D$ be a domain in $R^d$ and $u$ be the solution to the stochastic heat equation $$ du=Δu dt+ g\,dW_t, \quad t>0, x\in D, $$ with zero initial and boundary data. Here $W_t$ is a one-dimensional Wiener process on a probability space $Ω$. It has been proved (see below for references) that for any $p\geq 2$ the inequality $$ \|\nabla u\|_{L_p(Ω\times [0,T]\times D)} \leq c \|g\|_{L_p(Ω\times [0,T]\times D)} $$ holds if $\partial D\in C^1$. In this note we prove that if $p>4$ then this inequality fails in any polygon in $R^2$ having an angle greater than or equal to $\frac{pπ}{2(p-2)}$. We also show that a similar statement holds in higher dimensional polygons. The counterexample introduced here is based on personal communication with N.V. Krylov.
In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^α_tu=L(ω,t,x)u+f(u)+\partial^β_t \sum_{k=1}^{\infty}\int^t_0 ( Λ^k(ω,t,x)u+g^k(u))dw^k_t, $$ where $α\in (0,2)$, $β<α+\frac{1}{2}$, and $\partial^α_t$ and $\partial^β_t$ denote the Caputo derivative of order $α$ and $β$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $Ω$, $L$ is a second order operator of either divergence or non-divergence type, and $Λ^k$ are linear operators of order up to two. The coefficients of the equations depend on $ω(\in Ω), t,x$ and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^αu=-(-Δ)^βu, \quad α\in (0,2), \, β\in (0,\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-Δ_x)^γD_{t}^σI_{t}^δp(t,x), \quad \forall\,\, n\in\mathbb{Z}_{+}, \,\, γ\in[0,β],\,\, σ, δ\in[0,\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-Δ_x)^γ$ is a fractional Laplace operator and $D_{t}^σ$ and $I_{t}^δ$ are Riemann-Liouville fractional derivative and integral respectively.