arXiv subjects
Publications and source records attributed to Sujin Khomrutai.
We discuss tangential Sobolev-estimates up to the boundary for solutions to the regional fractional laplacian on the upper half-plane. These estimates can be used to reduce the boundary Calderon-Zygmund theory of any dimension to a one-dimensional nonlocal problem.
We show that the operator \[ T_{K,s_1,s_2}f(z) := \int_{\mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 \] is a Calderon-Zygmund operator. Here for $K \in L^\infty(\mathbb{R}^n \times \mathbb{R}^n)$, and $s,s_1,s_2 \in (0,1)$ with $s_1+s_2 = 2s$ we have \[ A_{K,s_1,s_2}(z_1,z_2) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) \left (|x-z_1|^{s_1-n} -|y-z_1|^{s_1-n} \right )\, \left (|x-z_2|^{s_2-n} -|y-z_2|^{s_2-n}\right )}{|x-y|^{n+2s}}\, dx\, dy. \] This operator is motivated by the recent work by Mengesha-Schikorra-Yeepo where it appeared as analogue of the Riesz transforms for the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = f[φ]. \]
We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash \mathbb{B}^N(0)$}. \] Our main result is $C^{1,α}$-regularity for any $α< 1$. Technically, we split the map $u=λv$, where $v: \mathbb{B}^2 \to \mathbb{S}^{N-1}$ is the vectorial component and $λ= |u|$ the scalar component measuring the distance to the origin. While $v$ satisfies a weighted harmonic map equation with weight $λ^2$, $λ$ solves the obstacle problem for \[ \int_{\mathbb{B}^2} |\nabla λ|^2+λ^2 |\nabla v|^2, \quad \mbox{subject to $λ\geq 1$}. \] where $|\nabla v|^2 \in L^1(\mathbb{B}^2)$. We then play ping-pong between the increases in the regularity of $λ$ and $v$ to obtain finally the $C^{1,α}$-result.
We investigate a nonlocal equation $\partial_tu=\int_{\mathbb{R}^n}J(x-y)(u(y,t)-u(x,t))dy+a(x,t)u^p$ in $\mathbb{R}^n$, where $a$ is unbounded and $J$ belongs to a weighted space. Crucial weighted $L^p$ and interpolation estimates for the Green operator are established by a new method based on the sharp Young's inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions $\varGamma=(1+|x|^2/η)^{b/2}$ that $-\varGamma/η\lesssim J\ast\varGamma-\varGamma\lesssim\varGamma/η$ and $η^{-b_+/2}\lesssim\varGamma/\left\langle x\right\rangle^b\lesssim η^{-b_-/2}$. Blow-up behaviors are investigated by employing test functions $ϕ_R=\varGamma$ ($η=R$) instead of principal eigenfunctions. Global well-posedness in weighted $L^p$ spaces for the Cauchy problem is proved. When $a\sim\left\langle x\right\rangle^σ$ the Fujita exponent is shown to be $1+(σ+2)/n$. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.
We study the asymptotic behavior for nonlocal diffusion equations $\partial_tu=\mathcal{J}u-χ_0u$ in $\mathbb{R}^n\times(0,\infty)$ and obtain a sufficient condition so that solutions of the Cauchy problem decay in time at the rate of a regular varying function. In the sufficient condition, a sharp bound of certain forms is required for the $k$-fold iterations $\mathcal{J}^ku_0$ or the kernels $J_k$. We prove the desired decay rate by analyzing the asymptotic behavior of a regular varying modified exponential series. Then we verify that the sufficient condition is true for most of the known radially symmetric kernels, and for some more general kernels, using the sharp Young's convolution inequality and a Fourier splitting argument. Classical results on the decay of solutions for these nonlocal diffusion equations are re-established and generalized. Finally, using our framework, we can exhibit a kernel having a prescribed regular varying decay solutions for a wide class of regular varying functions.
We study linear pseudoparabolic equations with unbounded and time-dependent coefficients. We solve the case which has remained open in several recent studies of pseudoparabolic equations with unbounded and time-dependent coefficients. In this work we get a sharp condition for the existence or non-existence of solutions. Conditions on the initial function and coefficient are provided so that every nontrivial positive solution blows up instantaneously. When the coefficient and the initial function do not grow too rapidly, we establish the existence and uniqueness of global solutions, for both time-independent and time-dependent potentials. This is done via the analysis of the Bessel convolution multiplication operators. Asymptotic behavior and comparison principles are also established. The global well-posedness results can be extended to the equation with convection.
We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is established within weighted continuous spaces by approximation and monotonicity arguments. Every solution with non-zero initial value is shown to exhibit a certain lower grow-up and radial growth bound, depending only upon the sublinearity and the unbounded, time-dependent potential. Using the lower grow-up/growth bound, we can prove the key comparison principle. Then we settle the uniqueness of solutions for the problem with non-zero initial condition by employing the comparison principle. For the problem with the zero initial condition, we can classify the non-trivial solutions in terms of the maximal solutions. When the initial condition has a power radial growth, we can derive the precise asymptotic grow-up rate of solutions and obtain the critical growth exponent.