Catharine W. K. Lo — Page 2

arXiv2022

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

Consider the quasilinear diffusion problem \[\begin{cases}\mathbf{u}'+Π(t,x,\mathbf{u},Σ\mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},Σ\mathbf{u})&\text{ in }]0,T[\timesΩ,\\\mathbf{u}=\mathbf{0}&\text{ in }]0,T[\timesΩ^c,\\\mathbf{u}(0,\cdot)=\mathbf{u}_0(\cdot)&\text{ in }Ω\end{cases}\] for an open set $Ω\subset\mathbb{R}^n$, $\mathbf{u}_0\in \mathbf{H}^s_0(Ω):=[H^s_0(Ω)]^m$ and any $T\in]0,\infty[$, where $Σ\mathbf{u}\in \mathbb{R}^q$ for $0<q\leq m\times n$ represents fractional or nonlocal derivatives with order $σ$ with $σ<2s$ for all $0<s\leq1$, including the classical gradient and derivatives of order greater than 1. We show global existence results for various quasilinear diffusion systems in non-divergence form, for different linear operators $\mathbb{A}$, including local elliptic systems, anisotropic fractional equations and systems, and anisotropic nonlocal operators, of the following type \[(\mathbb{A}\mathbf{u})^i=-\sum _{α,β,j} \partial_α(A^{αβ}_{ij}\partial_βu^j),\quad \mathbb{A}u=- D^s(A(x)D^su),\quad\text{ and }\quad (\mathbb{A}\mathbf{u})^i=\int_{\mathbb{R}^n}A_{ij}(x,y)\frac{u^j(x)-u^j(y)}{|x-y|^{n+2s}}\,dy,\] for coercive, invertible matrices $Π$ and suitable vectorial functions $\mathbf{f}$.

math.AP↗

arXiv2021

On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative

In this work, we consider the nonlocal obstacle problem with a given obstacle $ψ$ in a bounded Lipschitz domain $Ω$ in $\mathbb{R}^{d}$, such that $\mathbb{K}_ψ^s=\{v\in H^s_0(Ω):v\geqψ\text{ a.e. in }Ω\}\neq\emptyset$, given by \[u\in\mathbb{K}_ψ^s:\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_ψ,\] for $F\in H^{-s}(Ω)$, the dual space of $H^s_0(Ω)$, $0<s<1$. The nonlocal operator $\mathcal{L}_a:H^s_0(Ω)\to H^{-s}(Ω)$ is defined with a measurable, bounded, strictly positive singular kernel $a(x,y)$, possibly not symmetric, by \[\langle\mathcal{L}_au,v\rangle=P.V.\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}v(x)(u(x)-u(y))a(x,y)dydx=\mathcal{E}_a(u,v),\] with $\mathcal{E}_a$ being a Dirichlet form. Also, the fractional operator $\tilde{\mathcal{L}}_A=-D^s\cdot AD^s$ defined with the distributional Riesz $s$-fractional derivative and a bounded matrix $A(x)$ gives a well defined integral singular kernel. The corresponding $s$-fractional obstacle problem converges as $s\nearrow1$ to the obstacle problem in $H^1_0(Ω)$ with the operator $-D\cdot AD$ given with the gradient $D$. We mainly consider problems involving the bilinear form $\mathcal{E}_a$ with one or two obstacles, and the N-membranes problem, deriving a weak maximum principle, comparison properties, approximation by bounded penalization, and the Lewy-Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in $L^\infty(Ω)$, local Hölder regularity when $a$ is symmetric, and local regularity in $W^{2s,p}_{loc}(Ω)$ and $C^1(Ω)$ for fractional $s$-Laplacian obstacle-type problems. These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of $H^s_0(Ω)$ and some remarks on the associated $s$-capacity for general $\mathcal{L}_a$.

math.AP↗