arXiv subjects
Publications and source records attributed to Yeonghun Youn.
We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = μ\] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calderón--Zygmund estimates in the singular case $2-1/n < p < 2$, under natural assumptions on $p$, $q$ and $a(\cdot)$.
We study nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = μ, \] where $2\le p<q<\infty$, $a(\cdot) \ge 0$, and $μ$ is a signed Borel measure with finite total mass. We prove local Calderón--Zygmund type gradient estimates for SOLA (Solutions Obtained as Limits of Approximations) by finding new and natural assumptions on $p$, $q$ and $a(\cdot)$.
We study the existence of very weak solutions to a system \[\begin{cases}-\mathrm{div} \mathcal{A}(x,D\mathbf{u})=\mathbfμ\quad\text{in }\ Ω, \mathbf{u}=0\quad\text{on }\ \partialΩ\end{cases} \] with a datum $\mathbfμ$ being a vector-valued bounded Radon measure and $\mathcal{A}$ having measurable dependence on the spacial variable and Orlicz growth with respect to the second variable. We are {\em not} restricted to the superquadratic case. For the solutions and their gradients we provide regularity estimates in the generalized Marcinkiewicz scale. In addition, we show a precise sufficient condition for the solution to be a~Sobolev function.
We study gradient regularity for mixed local-nonlocal problems modelled upon \[ -Δ_p u +(-Δ_p)^su=μ\qquad\text{for} \quad 2-\tfrac{1}{n}<p<\infty\quad \text{and}\quad s\in(0,1)\,,\] where $μ$ is a bounded Borel measure. We prove pointwise bounds for the gradient $Du$ in terms of the truncated 1-Riesz potential of $μ$.
We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.
We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for $p$-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.